Linear Algebra
2.1.34 Give a formula for (ABx)T, where x is a vector and A and B are matrices of appropriate size. Choose the correct answer below. A. (ABx)T=BTATxT, because (ABx)T=(AB)TxT=BTATxT B. (ABx)T=xTBTAT, because (ABx)T=xT(AB)T=xTBTAT C. (ABx)T=ATBTxT, because (ABx)T=(AB)TxT=ATBTxT D. (ABx)T=xTATBT, because (ABx)T=xT(AB)T=xTAT
(ABx)T=xTBTAT, because (ABx)T=xT(AB)T=xTBTAT
How many rows does B have if BC is a 4×9 matrix? Matrix B has _?_ rows.
4
2.1.7 If a matrix A is 4×5 and the product AB is 4×3, what is the size of B? The size of B is _?_×_?_.
5 x 3
2.1.19 If A, B, and C are n×n invertible matrices, does the equation C−1(A+X)B−1=In have a solution X? If so, find it. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The solution is X=_?_. B. There is no solution.
A. CB-A
2.3.20 If the n×n matrices E and F have the property that EF=I, then E and F commute. Explain why. Select the correct choice below.| A. According the Invertible Matrix Theorem, E and F must not be invertible and therefore cannot be inverses. So FE=I and I=EF. Thus, E and F commute. B. According the Invertible Matrix Theorem, E and F must be invertible and inverses. So FE=I and I=EF. Thus, E and F commute. C. According the Invertible Matrix Theorem, E and F must both be identity matrices. So F=I and E=I; therefore, FE=EF. Thus, E and F commute. D. According the Invertible Matrix Theorem, E and F both have columns that span ℝn. So FE=EF. Thus, E and F commute.
According the Invertible Matrix Theorem, E and F must be invertible and inverses. So FE=I and I=EF. Thus, E and F commute.
2.1.17 Solve the equation AB=BC for A, assuming that A, B, and C are square matrices and B is invertible. A=_?_
BCB^-1
2.3.19 If the columns of a 7×7 matrix D are linearly independent, what can you say about the solutions of Dx=b? Why? Select the correct choice below. A. It will depend on the values in the matrix. If the diagonal of the matrix is zero, Dx=b has a solution for each b in ℝ7. However, if the diagonal is all non-zero, equation Dx=b has many solutions for each b in ℝ7. B. Equation Dx=b has no solutions for each b in ℝ7. According to the Invertible Matrix Theorem, the equation Dx=0 has only the trivial solution. C. Equation Dx=b has a solution for each b in ℝ7. According to the Invertible Matrix Theorem, a matrix is invertible if the columns of the matrix form a linearly independent set; this would mean that the equation Dx=b has at least one solution for each b in ℝn. D. Equation Dx=b has many solutions for each b in ℝ7. According to the Invertible Matrix Theorem, a matrix is not invertible if the columns of the matrix form a linearly independent set, and the equation Dx=b has many solutions for each b in ℝn.
Equation Dx=b has a solution for each b in ℝ7. According to the Invertible Matrix Theorem, a matrix is invertible if the columns of the matrix form a linearly independent set; this would mean that the equation Dx=b has at least one solution for each b in ℝn.
1.8.31 Let T: ℝn→ℝm be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in ℝn. Explain why the set {T(v1), T(v2), T(v3)} is linearly dependent. Choose the correct answer below. A. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0. It follows that c1T(v1)+c2T(v2)+c3T(v3)≠0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. B. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0. It follows that c1T(v1)+c2T(v2)+c3T(v3)=0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. C. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3≠0. It follows that c1T(v1)+c2T(v2)+c3T(v3)≠0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. D. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, all zero, such that c1v1+c2v2+c3v3=0. It follows that c1T(v1)+c2T(v2)+c3T(v3)=0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent.
Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0. It follows that c1T(v1)+c2T(v2)+c3T(v3)=0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent.
2.2.21 Explain why the columns of an n×n matrix A are linearly independent when A is invertible. Choose the correct answer below. A. If A is invertible, then A has an inverse matrix A−1. Since AA−1=I, A must have linearly independent columns. B. If A is invertible, then A has an inverse matrix A−1. Since AA−1=A−1A, A must have linearly independent columns. C. If A is invertible, then for all x there is a b such that Ax=b. Since x=0 is a solution of Ax=0, the columns of A must be linearly independent. D. If A is invertible, then the equation Ax=0 has the unique solution x=0. Since Ax=0 has only the trivial solution, the columns of A must be linearly independent.
If A is invertible, then the equation Ax=0 has the unique solution x=0. Since Ax=0 has only the trivial solution, the columns of A must be linearly independent.
2.3.26 Explain why the columns of A2 span ℝn whenever the columns of an n×n matrix A are linearly independent. Choose the correct answer below. Note that the invertible matrix theorem is abbreviated IMT. A. If the columns of A are linearly independent, then it directly follows that the columns of A2 span ℝn. B. If the columns of A are linearly independent and A is square, then A is invertible, by the IMT. Thus, A2, which is the product of invertible matrices, is not invertible. So, the columns of A2 span ℝn. C. If the columns of A are linearly independent and A is square, then A is not invertible. Thus, A2, which is the product of non invertible matrices, is also not invertible. So, the columns of A2 span ℝn. D. If the columns of A are linearly independent and A is square, then A is invertible, by the IMT. Thus, A2, which is the product of invertible matrices, is also invertible. So, by the IMT, the columns of A2 span ℝn.
If the columns of A are linearly independent and A is square, then A is invertible, by the IMT. Thus, A2, which is the product of invertible matrices, is also invertible. So, by the IMT, the columns of A2 span ℝn.
2.2.24 Suppose A is n×n and the equation Ax=b has a solution for each b in ℝn. Explain why A must be invertible. [Hint: Is A row equivalent to In?] Choose the correct answer below. A. If the equation Ax=b has a solution for each b in ℝn, then A does not have a pivot position in each row. Since A is square, and In is square, A is row equivalent to In. Therefore, A is invertible. B. If the equation Ax=b has a solution for each b in ℝn, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A. It follows that A is In. Therefore, A is invertible. C. If the equation Ax=b has a solution for each b in ℝn, then A has one pivot position. It follows that A is row equivalent to In. Therefore, A is invertible. D. If the equation Ax=b has a solution for each b in ℝn, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A. It follows that A is row equivalent to In. Therefore, A is invertible.
If the equation Ax=b has a solution for each b in ℝn, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A. It follows that A is row equivalent to In. Therefore, A is invertible.
2.3.17 If A is invertible, then the columns of A−1 are linearly independent. Explain why. Select the correct choice below. A. According to the Invertible Matrix Theorem, if a matrix is invertible its columns form a linearly dependent set. When the columns of a matrix are linearly dependent, then the columns of the inverse of that matrix are linearly independent. Therefore, the columns of A−1 are linearly independent. B. It is a known theorem that if A is invertible then A−1 must also be invertible. According to the Invertible Matrix Theorem, if a matrix is invertible its columns form a linearly independent set. Therefore, the columns of A−1 are linearly independent. C. The columns of A−1 are linearly independent because A is a square matrix, and according to the Invertible Matrix Theorem, if a matrix is square, it is invertible and its columns are linearly independent. D. If A is invertible, then the rows of A are linearly independent, which implies that the columns of A−1 are linearly independent.
It is a known theorem that if A is invertible then A−1 must also be invertible. According to the Invertible Matrix Theorem, if a matrix is invertible its columns form a linearly independent set. Therefore, the columns of A−1 are linearly independent.
2.3.18 If C is 6×6 and the equation Cx=v is consistent for every v in ℝ6, is it possible that for some v, the equation Cx=v has more than one solution? Why or why not? Select the correct choice below. A. It is possible because 6×6 is a square matrix, and according to the Invertible Matrix Theorem all square matrices are not invertible. Since it is not invertible, Cx=v does not have a unique solution. B. It is possible. Since Cx=v is consistent for every v in ℝ6, according to the Invertible Matrix Theorem that makes the 6×6 matrix not invertible. Since it is not invertible, Cx=v does not have a unique solution. C. It will depend on the values in the 6×6 matrix. According to the Invertible Matrix Theorem if any of the values are zero this makes Cx=v have more than one solution. If none of the values are zero, then Cx=v has a unique solution. D. It is not possible. Since Cx=v is consistent for every v in ℝ6, according to the Invertible Matrix Theorem that makes the 6×6 matrix invertible. Since it is invertible, Cx=v has a unique solution.
It is not possible. Since Cx=v is consistent for every v in ℝ6, according to the Invertible Matrix Theorem that makes the 6×6 matrix invertible. Since it is invertible, Cx=v has a unique solution.
2.3.16 Is it possible for a 5×5 matrix to be invertible when its columns do not span ℝ5? Why or why not? Select the correct choice below. A. It is not possible; according to the Invertible Matrix Theorem an n×n matrix cannot be invertible when its columns do not span ℝn. B. It is possible; according to the Invertible Matrix Theorem all square matrices are always invertible. C. It will depend on the values in the matrix. According to the Invertible Matrix Theorem, a square matrix is only invertible if it is row equivalent to the identity. D. It is possible; according to the Invertible Matrix Theorem an n×n matrix can be invertible when its columns do not span ℝn.
It is not possible; according to the Invertible Matrix Theorem an n×n matrix cannot be invertible when its columns do not span ℝn.
2.2.8 Use matrix algebra to show that if A is invertible and D satisfies AD=I, then D=A−1. Choose the correct answer below. A. Right-multiply each side of the equation AD=I by A−1 to obtain ADA−1=IA−1, DI=A−1, and D=A−1. B. Add A−1 to both sides of the equation AD=I to obtain A−1+AD=A−1+I, ID=A−1, and D=A−1. C. Add A−1 to both sides of the equation AD=I to obtain AD+A−1=I+A−1, DI=A−1, and D=A−1. D. Left-multiply each side of the equation AD=I by A−1 to obtain A−1AD=A−1I, ID=A−1, and D=A−1.
Left-multiply each side of the equation AD=I by A−1 to obtain A−1AD=A−1I, ID=A−1, and D=A−1.
2.3.28 Let A and B be n×n matrices. Show that if AB is invertible so is B. Choose the correct answer below. Note that the invertible matrix theorem is abbreviated IMT. A. Since AB is invertible, then it directly follows that A=B−1 and B=A−1 by the IMT. Therefore, matrix B is invertible. B. Let W be the inverse of AB. Then WAB=I and (WA)B=I. Therefore, matrix B is invertible by part (j) of the IMT. C. Since AB is invertible then by the IMT ABT is an invertible matrix. Therefore, matrix B is invertible by part (l) of the IMT. D. Let W be the inverse of AB. Then WAB=B. Therefore, since B is the product of two invertible matrices, W and AB, matrix B is invertible.
Let W be the inverse of AB. Then WAB=I and (WA)B=I. Therefore, matrix B is invertible by part (j) of the IMT.
1.5.28 If b≠0, can the solution set of Ax=b be a plane through the origin? Choose the correct answer. A. No. If the solution set of Ax=b contained the origin, then 0 would satisfy A0=b, which is not true since b is not the zero vector. B. Yes. Since the solution set of Ax=0 contains the origin, the solution set of Ax=b must contain the origin. C. Yes. The solution set of Ax=b is always represented as a plane through the origin. D. No. The solution set of Ax=b contains the origin if and only if Ax=b is inconsistent, which is not true for any particular vector b.
No. If the solution set of Ax=b contained the origin, then 0 would satisfy A0=b, which is not true since b is not the zero vector.
1.5.38 Suppose A is a 3×3 matrix and y is a vector in ℝ3 such that the equation Ax=y does not have a solution. Does there exist a vector z in ℝ3 such that the equation Ax=z has a unique solution? Choose the correct answer. A. No. Since Ax=y has no solution, then A cannot have a pivot in every row. So the equation Ax=z has at most two basic variables and at least one free variable for any z. Thus the solution set for Ax=z is either empty or has infinitely many elements. B. No. Since Ax=y has no solution, then A cannot have a pivot in every row. Since A is 3×3, it has at most two pivot positions. So the equation Ax=z has at most two basic variables and at least one free variable for any z. Thus there is no solution for Ax=z. C. Yes. Since Ax=y has no solution, then A cannot have a pivot in every row. So there is at least one free variable. The free variable(s) can be set equal to values such that there is a unique solution for Ax=z for any given z. D. Yes. Since Ax=y has no solution, then A cannot have a pivot in every row. The free variable(s) can be set equal to values such that there are infinitely many solutions for Ax=z for any given z.
No. Since Ax=y has no solution, then A cannot have a pivot in every row. So the equation Ax=z has at most two basic variables and at least one free variable for any z. Thus the solution set for Ax=z is either empty or has infinitely many elements.
2.2.22 Explain why the columns of an n×n matrix A span ℝn when A is invertible. Choose the correct answer below. A. Since A is invertible, there exists A−1 such that AA−1=I. Since AA−1=I, the columns of A span ℝn. B. Since A is invertible, for each b in ℝn the equation Ax=b has a unique solution. Since the equation Ax=b has a solution for all b in ℝn, the columns of A span ℝn. C. Since A is invertible, each b is a linear combination of the columns of A. Since each b is a linear combination of the columns of A, the columns of A span ℝn. D. Since A is invertible, det A is zero. Since det A is zero, the columns of A span ℝn.
Since A is invertible, for each b in ℝn the equation Ax=b has a unique solution. Since the equation Ax=b has a solution for all b in ℝn, the columns of A span ℝn.
1.5.26 Suppose Ax=b has a solution. Explain why the solution is unique precisely when Ax=0 has only the trivial solution. Choose the correct answer. A. Since Ax=b is inconsistent, its solution set is obtained by translating the solution set of Ax=0. For Ax=b to be inconsistent, Ax=0 has only the trivial solution. B. Since Ax=b is inconsistent, then the solution set of Ax=0 is also inconsistent. The solution set of Ax=0 is inconsistent if and only if Ax=0 has only the trivial solution. C. Since Ax=b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax=0 has only the trivial solution. D. Since Ax=b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax=b is a single vector if and only if the solution set of Ax=0 is a single vector, and that happens if and only if Ax=0 has only the trivial solution.
Since Ax=b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax=b is a single vector if and only if the solution set of Ax=0 is a single vector, and that happens if and only if Ax=0 has only the trivial solution.
2.2.23 Suppose A is n×n and the equation Ax=0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In. Choose the correct answer below. A. Suppose A is n×n and the equation Ax=0 has only the trivial solution. Then there are n free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the n×n identity matrix, In. B. Suppose A is n×n and the equation Ax=0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in A must be on the main diagonal. Hence A is the n×n identity matrix, In. C. Suppose A is n×n and the equation Ax=0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the n×n identity matrix, In.
Suppose A is n×n and the equation Ax=0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the n×n identity matrix, In.
2.1.21 Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you say about the columns of A? Choose the correct answer below. A. The columns of A are linearly dependent because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. Since bp is not all zeros, then any solution to Abp=0 can not be the trivial solution. B. The columns of A must all have entries of zero because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. This implies that A must have columns that are composed of only zeros and therefore AB must be the zero matrix. C. The columns of A must be the identity matrix columns because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. This implies that A must be the identity matrix to ensure that the last column of AB is all zeros. D. The columns of A are linearly independent because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. Since bp is not all zeros, then the only solution to Abp=0 is the trivial solution.
The columns of A are linearly dependent because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. Since bp is not all zeros, then any solution to Abp=0 can not be the trivial solution.
2.3.21 If the given equation Gx=y has more than one solution for some y in ℝn, can the columns of G span ℝn? Why or why not? Assume G is n×n. Select the correct choice below. A. The columns of G can span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, that makes the matrix G invertible. B. The columns of G cannot span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, that makes the matrix G non invertible. C. The columns of G cannot span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, the columns of A form a linearly independent set. D. `The columns of G can span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, the transformation x ↦ Ax is one-to-one.
The columns of G cannot span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, that makes the matrix G non invertible.
2.3.23 If an n×n matrix K cannot be row reduced to In, what can you say about the columns of K? Why? Select the correct choice below. A. The columns of K are linearly independent and the columns span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In that matrix is invertible. B. The columns of K are linearly independent and the columns span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In, the equation Ax=b has at least one solution for each b in ℝn. C. The columns of K are linearly dependent and the columns do not span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In that matrix is non invertible. D. The columns of K are linearly dependent and the columns span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In, the linear transformation x ↦ Ax is one-to-one.
The columns of K are linearly dependent and the columns do not span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In that matrix is non invertible.
1.4.33 Suppose A is a 4×3 matrix and b is a vector in ℝ4 with the property that Ax=b has a unique solution. What can you say about the reduced echelon form of A? Justify your answer. Choose the correct answer below. A. The first term of the first row will be a 1 and all other terms will be 0. There is only one variable xm, so there is only one possible solution. B. There will be a pivot position in each row. If a row did not have a pivot position then the equation Ax=b would be inconsistent. C. The first row will have a pivot position and all other rows will be all zeros. There is only one equation to solve, so there is only one solution. D. The first 3 rows will have a pivot position and the last row will be all zeros. If a row had more than one 1, then there would be an infinite number of solutions for amxm=bm.
The first 3 rows will have a pivot position and the last row will be all zeros. If a row had more than one 1, then there would be an infinite number of solutions for amxm=bm.
2.1.18 Suppose the first two columns, b1 and b2, of B are equal. What can you say about the columns of AB (if AB is defined)? Why? Choose the correct answer below. A. The first two columns of AB will be equal only if the first two rows of A are equal. B. Nothing can be determined about the columns of AB since the entries of A are unknown. C. The first two columns of AB will be equal only if the first two columns of A are equal. D. The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.
The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.
1.8.8 How many rows and columns must a matrix A have in order to define a mapping from ℝ5 into ℝ8 by the rule T(x)=Ax? Choose the correct answer below. A. The matrix A must have 8 rows and 8 columns. B. The matrix A must have 5 rows and 5 columns. C. The matrix A must have 8 rows and 5 columns. D. The matrix A must have 5 rows and 8 columns.
The matrix A must have 8 rows and 5 columns.
1.1.17 Do the three lines 2x1−4x2=14, 4x1+6x2=−56, and −2x1−10x2=70 have a common point of intersection? Explain. Choose the correct answer below. A. The three lines do not have a common point of intersection. B. The three lines have at least one common point of intersection. C. There is not enough information to determine whether the three lines have a common point of intersection.
The three lines have at least one common point of intersection.
2.3.15 Can a square matrix with two identical columns be invertible? Why or why not? Select the correct choice below. A. The matrix is invertible. If a matrix has two identical columns then its columns are linearly independent. According to the Invertible Matrix Theorem this makes the matrix invertible. B. The matrix is not invertible. If a matrix has two identical columns then its columns are linearly dependent. According to the Invertible Matrix Theorem this makes the matrix not invertible. C. It depends on the values in the matrix. According to the Invertible Matrix Theorem, if the two columns are larger than any other columns the matrix will be invertible, otherwise it will not. D. The matrix is not invertible. According to the Invertible Matrix Theorem a square matrix can never be invertible.
The matrix is not invertible. If a matrix has two identical columns then its columns are linearly dependent. According to the Invertible Matrix Theorem this makes the matrix not invertible.
1.4.34 Suppose A is a 3×3 matrix and b is a vector in ℝ3 with the property what Ax=b has a unique solution. Explain why the columns of A must span ℝ3. Choose the correct answer below. A. When b is written as a linear combination of the columns of A, it simplifies to the vector of weights, x. Therefore the columns of A must span ℝ3. B. The equation has a unique solution so for each pair of vectors x and b there is only one possible matrix A. Therefore the columns of A must span ℝ3. C. Matrix A is a square matrix, so when computing Ax, the row-vector rule shows that the columns of A must span ℝ3. D. The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation Ax=b. Therefore the columns of A must span ℝ3.
The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation Ax=b. Therefore the columns of A must span ℝ3.
2.1.19 Suppose the sixth column of B is the sum of the first two columns. What can be said about the sixth column of AB? Why? What can be said about the sixth column of AB? Why? A. The sixth column of AB is the sum of the first two columns of AB. If B is b1b2...bp, then the sixth column of AB is Ab6 by definition. It is given that b6=b1+b2. By matrix-vector multiplication, Ab6=Ab1+b2=Ab1+Ab2. B. The sixth column of AB is the sum of the first two columns of B. If B is b1b2...bp, then the sixth column of AB is Ab6 by definition. It is given that b6=b1+b2. By matrix-vector multiplication, Ab6=b1+b2=b1+b2. C. The sixth column of AB is the sum of the first two columns of AB. If B is b1b2...bp, then the sixth column of AB is Ab6 by definition. It is given that b6=b1−b2. By matrix-vector multiplication, Ab6=Ab1−b2=Ab1−Ab2. D. The sixth column of AB is the sum of the first two columns of B. If B is b1b2...bp, then the sixth column of AB is Ab6 by definition. It is given that b6=b1−b2. By matrix-vector multiplication,
The sixth column of AB is the sum of the first two columns of AB. If B is b1b2...bp, then the sixth column of AB is Ab6 by definition. It is given that b6=b1+b2. By matrix-vector multiplication, Ab6=Ab1+b2=Ab1+Ab2.
1.7.35 The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. If v1 and v2 are in ℝ4 and v2 is not a scalar multiple of v1, then {v1,v2} is linearly independent. Choose the correct answer below. A. The statement is true. A set of vectors is linearly independent if and only if none of the vectors are a scalar multiple of another vector. B. The statement is false. The vector v1 could be the zero vector. C. The statement is false. The vector v1 could be equal to the vector v2. D. The statement is false. The vector v1 could be a scalar multiple of vector v2.
The statement is false. The vector v1 could be the zero vector.
2.3.22 Suppose H is an n×n matrix. If the equation Hx=c is inconsistent for some c in ℝn, what can you say about the equation Hx=0? Why? Select the correct choice below. A. The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has no solution for some c. From this, all of the statements in the Invertible Matrix Theorem are false, including the statement that Hx=0 has only the trivial solution. Thus, Hx=0 has no solution. B. The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has a solution for every c. From this, all of the statements in the Invertible Matrix Theorem are true, including the statement that the columns of H form a linearly independent set. Thus, Hx=0 has an infinite number of solutions. C. The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has a solution for every c. From this, all of the statements in the Invertible Matrix Theorem are true, including the statement that Hx=0 has only the trivial solution. D. The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has no solution for some c. From this, all of the statements in the Invertible Matrix Theorem are false, including the statement that Hx=0 has only the trivial solution. Thus, Hx=0 has a nontrivial solution.
The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has no solution for some c. From this, all of the statements in the Invertible Matrix Theorem are false, including the statement that Hx=0 has only the trivial solution. Thus, Hx=0 has a nontrivial solution.
1.2.25 Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent. Choose the correct answer below. A. The system is consistent because the augmented matrix is row equivalent to one and only one reduced echelon matrix. B. The system is consistent because the augmented matrix will contain a row of the form 0...0b with b nonzero. C. The system is consistent because all the columns in the augmented matrix will have a pivot position. D. The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
1.1.18 Do the three planes x1+4x2+2x3=5, x2−2x3=1, and 3x1+15x2=15 have at least one common point of intersection? Explain. Choose the correct answer below. A. The three planes do not have a common point of intersection. B. The three planes have at least one common point of intersection. C. There is not enough information to determine whether the three planes have a common point of intersection.
The three planes do not have a common point of intersection.
1.2.23 Suppose a 6×8 coefficient matrix for a system has six pivot columns. Is the system consistent? Why or why not? Choose the correct answer below. A. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have seven columns and will not have a row of the form [0000001], so the system is consistent. B. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have nine columns, must have a row of the form [000000001], so the system is inconsistent. C. There is a pivot position in each row of the coefficient matrix. The augmented matrix will have nine columns and will not have a row of the form [000000001], so the system is consistent. D. There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have nine columns, could have a row of the form [000000001], so the system could be inconsistent.
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have nine columns and will not have a row of the form [000000001], so the system is consistent.
If L is n×n and the equation Lx=0 has the trivial solution, do the columns of L span ℝn? Why? Select the correct choice below. A. If Lx=0 has the trivial solution and L is square, then L is invertible, and the columns of L span ℝn. B. If Lx=0 has the trivial solution, then according to the Invertible Matrix Theorem, Lx=b has a solution for all b and the columns of L span ℝn. C. The columns of L do not span ℝn. If Lx=0 has the trivial solution, then according to the Invertible Matrix Theorem, L is not invertible and the columns of L do not span ℝn. D. This fact gives no information about the columns of L. The equation Lx=0 always has the trivial solution.
This fact gives no information about the columns of L. The equation Lx=0 always has the trivial solution.
1.7.33 The statement is either true in all cases or false. If false, construct a specific example to show that the statement is not always true. If v1, ..., v4 are in ℝ4 and v3=2v1+v2, then {v1, v2, v3, v4} is linearly dependent. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. True. The vector v3 is a linear combination of v1 and v2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent. B. True. Because v3=2v1+v2, v4 must be the zero vector. Thus, the set of vectors is linearly dependent. C. True. If c1=2, c2=1, c3=1, and c4=0, then c1v1+•••+c4v4=0. The set of vectors is linearly dependent. D. False. If v1=nothing, v2=nothing, v3=nothing, and v4=1212, then v3=2v1+v2 and {v1, v2, v3, v4} is linearly independent.
True. The vector v3 is a linear combination of v1 and v2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.
2.3.37 Suppose T and U are linear transformations from ℝn to ℝn such that T(Ux)=x for all x in ℝn. Is it true that U(Tx)=x for all x in ℝn? Why or why not? Let A be the standard matrix for the linear transformation T and B be the standard matrix for the linear transformation U. Choose the correct answer below. A. Yes, it is true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x))=x is the trivial mapping and so AB=0. This implies that either A or B is the zero matrix, and so BA=0. This implies that U(T(x)) is also the trivial mapping. B. No, it is not true. ABT is the standard matrix for T(U(x)). By hypothesis, T(U(x))=x is the identity mapping and so ABT=I. However, this does not imply that BAT=I, where BAT is the standard matrix for U(T(x)). So U(T(x)) is not necessarily the identity matrix. C. No, it is not true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x))=x is the identity mapping and so AB=I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(T(x)), U(T(x)) is not necessarily the identity matrix. D. Yes, it is true. AB is the standard matrix of the mapping x↦T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB=I. Since both A and B are square and AB=I, the Invertible Matrix Theorem states that both A and B invertible, and B=A−1. Thus, BA=I. This means that the mapping x↦U(T(x)) is the identity mapping. Therefore, U(T(x))=x for all x in ℝn.
Yes, it is true. AB is the standard matrix of the mapping x↦T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB=I. Since both A and B are square and AB=I, the Invertible Matrix Theorem states that both A and B invertible, and B=A−1. Thus, BA=I. This means that the mapping x↦U(T(x)) is the identity mapping. Therefore, U(T(x))=x for all x in ℝn.
1.7.36 The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification. If v1, v2, v3 are in ℝ3 and v3 is not a linear combination of v1, v2, then {v1, v2, v3} is linearly independent. Fill in the blanks below. The statement is _?_ Take v1 and v2 to be multiples of one vector and take v3 to be not a multiple of that vector. For example, v1=111, v2=222, v3=100. Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly _?_ Choices: True, False Independent, Dependent
a) False b) Dependent
3.1.40 Let A be an n×n matrix. Mark each statement True or False. Justify each answer. a. The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row. b. The determinant of a triangular matrix is the sum of the entries on the main diagonal. a. Choose the correct answer below. A. True, because cofactor expansion across a row adds each of the cofactors together. Cofactor expansion down a column subtracts each cofactor from one another. This causes the two cofactor expansions to have opposite signs. B. True, because the plus or minus sign of the (i,j)-cofactor depends on the position of aij in matrix A. Cofactor expansion down a column switches the order of i and j, thereby switching the sign of the cofactor expansion across a row. C. False, because the determinant of A can only be calculated by cofactor expansion across a row. Cofactor expansion down a column has no relation to the determinant. D. False, because the determinant of A can be computed by cofactor expansion across any row or down any column. Since the determinant of A is well defined, both of these cofactor expansions will be equal. b. Choose the correct answer below. A. False, because the determinant of a triangular matrix is the product of the entries along the main diagonal. B. False, because the determinant of a matrix is the arithmetic mean of the entries along the main diagonal. C. True, because cofactor expansion along the row (or column) with the most zeros of a triangular matrix produces a determinant equal to the sum of the entries along the main diagonal. D. True, because the determinant of A is the following finite series. det A=∑j=1n(−1)1+ja1jdet A1j In a triangular matrix, this series simplifies to the sum of the entries along the main diagonal.
a) False, because the determinant of A can be computed by cofactor expansion across any row or down any column. Since the determinant of A is well defined, both of these cofactor expansions will be equal. b) False, because the determinant of a triangular matrix is the product of the entries along the main diagonal.
1.5.24 a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. A. True. A homogeneous equation cannot be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 does not have the solution x=0. Thus, a homogeneous system of equations can be inconsistent. B. True. A homogeneous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations can be inconsistent. C. False. A homogeneous equation cannot be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 does not have the solution x=0. Thus, a homogeneous system of equations cannot be inconsistent. D. False. A homogeneous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations cannot be inconsistent. b. If x is a nontrivial solution of Ax=0, then every entry in x is nonzero. Choose the correct answer below. A. False. A nontrivial solution of Ax=0 is the zero vector. Thus, a nontrivial solution x must have all zero entries. B. True. A nontrivial solution of Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero. C. True. A nontrivial solution of Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x cannot have any zero entries. D. False. A nontrivial solution of Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero. c. The effect of adding p to a vector is to move the vector in a direction parallel to p. Choose the correct answer below. A. False. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the line through v and 0. B. False. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the plane through v and 0. C. True. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the line through p and 0. D. False. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the plane through p and 0. d. The equation Ax=b is homogeneous if the zero vector is a solution. Choose the correct answer below. A. True. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a solution, then b=Ax=A0=0. B. False. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an m×n matrix and b is a nonzero vector in ℝm. Thus, the zero vector is never a solution of a homogeneous system. C. True. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an m×n matrix and b is a nonzero vector in ℝm. If the zero vector is a solution, then b=0. D. False. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a solution, then b=Ax=A0=0, which is false. e. If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0. Choose the correct answer below. A. True. Suppose the equation Ax=b is consistent for some given b. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is not a solution of the homogeneous equation Ax=0. B. True. Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the homogeneous equation Ax=0. C. False. Suppose the equation Ax=b is consistent for some given b. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is not a solution of the homogeneous equation Ax=0. D. False. Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the homogeneous equation Ax=0.
a) False. A homogeneous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely x=0. Thus, a homogeneous system of equations cannot be inconsistent. b) False. A nontrivial solution of Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero. c) True. Given v and p in ℝ2 or ℝ3, the effect of adding p to v is to move v in a direction parallel to the line through p and 0. d) True. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a solution, then b=Ax=A0=0. e) True. Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the homogeneous equation Ax=0.
1.7.21 In parts (a) to (d) below, mark the statement True or False. a. The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution. Choose the correct answer below. A. False. The columns of a matrix A are linearly independent only if the matrix equation Ax=0 has some solution other than the trivial solution. B. True. If a matrix equation has the trivial solution then there do not exist nonzero weights for the columns of A such that c1a1+c2a2+•••+cpap=0. C. False. For every matrix A, Ax=0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution. D. True. If the columns are linearly independent, then Ax=0 has the trivial solution. b. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. Choose the correct answer below. A. False. If an indexed set of vectors, S, is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set. B. True. If an indexed set of vectors, S, is linearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. Using the basic properties of equality, each of the vectors in the linear combination can also be written as a linear combination of those vectors. C. False. If S is linearly dependent then there is at least one vector that is not a linear combination of the other vectors, but the others may be linear combinations of each other. D. True. If S is linearly dependent then for each j, vj, a vector in S, is a linear combination of the preceding vectors in S. c. The columns of any 4×5 matrix are linearly dependent. Choose the correct answer below. A. True. When a 4×5 matrix is written in reduced echelon form, there will be at least one row of zeros, so the columns of the matrix are linearly dependent. B. False. If A is a 4×5 matrix then the matrix equation Ax=0 is inconsistent because the reduced echelon augmented matrix has a row with all zeros except in the last column. C. False. If a matrix has more rows than columns then the columns of the matrix are linearly dependent. D. True. A 4×5 matrix has more columns than rows, and if a set contains more vectors than there are entries in each vector, then the set is linearly dependent. d. If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}. Choose the correct answer below. A. True. If {x, y, z} is linearly dependent, then z must be a linear combination of x and y because x and y are linearly independent. So z is in Span{x, y}. B. False. Vector z cannot be in Span{x, y} because x and y are linearly independent. C. True. If {x, y, z} is linearly dependent and x and y are linearly independent, then z must be the zero vector. So z is in Span{x, y}. D. False. If x and y are linearly independent, and {x, y, z} is linearly dependent, then z must be the zero vector. So z cannot be in Span{x, y}.
a) False. For every matrix A, Ax=0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution. b) False. If an indexed set of vectors, S, is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set. c) True. A 4×5 matrix has more columns than rows, and if a set contains more vectors than there are entries in each vector, then the set is linearly dependent. d) True. If {x, y, z} is linearly dependent, then z must be a linear combination of x and y because x and y are linearly independent. So z is in Span{x, y}.
3.2.28 Mark each statement True or False. Justify each answer. Complete parts (a) through (d) below. a. If three row interchanges are made in succession, then the new determinant equals the old determinant. A. False. If three row interchanges are made in succession, then the new determinant equals 13 the old determinant. B. False. If three row interchanges are made in succession, then the new determinant equals the negative of the old determinant. C. True. If n row interchanges happen, then the determinant must be multiplied by (−1)n−1. D. True. Row interchanges do not affect the determinant. b. The determinant of A is the product of the diagonal entries in A. A. False. This is only true if A is triangular. B. True. The determinant is the product of the pivots and the pivots can be found on the diagonal. C. True. The determinnant of A is equal to the determinant of the reduced echelon form of A, and the determinant of the reduced echelon form of A is the product of its diagonal entries. D. False. The determinant is equal to ad−bc. c. If det A is zero, then two rows or two columns are the same, or a row or a column is zero. A. False. The determinant depends on the columns of A. It is possible for two rows to be the same and for the determinant to be nonzero. B. True. If det A is zero, then the columns of A are linearly independent. If one column is zero, or two columns are the same, then the columns are linearly dependent. C. False. If A=2613, then det A=0 and the rows and columns are all distinct and not full of zeros. D. True. If A=2323 and B=1200, then det A=0 and det B=0. d. det A−1=(−1)det A A. False. det A−1=det A B. False. det A−1=(det A)−1 C. True. det A−1 is the opposite of det A. D. True. A−1 is formed from an odd number of row or columns switches.
a) False. If three row interchanges are made in succession, then the new determinant equals the negative of the old determinant. b) False. This is only true if A is triangular. c) False. If A=2613, then det A=0 and the rows and columns are all distinct and not full of zeros. d) False. det A−1=(det A)−1
1.4.23 Determine whether each statement below is true or false. Justify each answer. a. The equation Ax=b is referred to as a vector equation. Choose the correct answer below. A. True. The equation Ax=b is referred to as a vector equation because A is constructed from column vectors. B. False. The equation Ax=b is referred to as a linear equation because b is a linear combination of vectors. C. True. The equation Ax=b is referred to as a vector equation because it consists of scalars multiplied by vectors. D. False. The equation Ax=b is referred to as a matrix equation because A is a matrix. b. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. Choose the correct answer below. A. True. The equation Ax=b is unrelated to whether the vector b is a linear combination of the columns of a matrix A. B. False. If the matrix A is the identity matrix, then the equation Ax=b has at least one solution, but b is not a linear combination of the columns of A. C. True. The equation Ax=b has the same solution set as the equation x1a1+x2a2+•••+x_na_n=b. Your answer is correct. D. False. If the equation Ax=b has infinitely many solutions, then the vector b cannot be a linear combination of the columns of A. c. The equation Ax=b is consistent if the augmented matrix Ab has a pivot position in every row. Choose the correct answer below. A. False. The augmented matrix Ab cannot have a pivot position in every row because it has more columns than rows. B. False. If the augmented matrix Ab has a pivot position in every row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b. Your answer is correct. C. True. The pivot positions in the augmented matrix Ab always occur in the columns that represent A. D. True. If the augmented matrix Ab has a pivot position in every row, then the equation Ax=b has a solution for each b in ℝm. d. The first entry in the product Ax is a sum of products. Choose the correct answer below. A. False. The first entry in Ax is the sum of the corresponding entries in x and the first entry in each column of A. B. True. The first entry in Ax is the sum of the products of corresponding entries in x and the first column of A. C. False. The first entry in Ax is the product of x1 and the column a1. D. True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A. Your answer is correct. e. If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm. Choose the correct answer below. A. True. Since the columns of A span ℝm, the augmented matrix Ab has a pivot position in every row. B. False. If the columns of A span ℝm, then the equation Ax=b is inconsistent for each b in ℝm. C. True. If the columns of A span ℝm, then the equation Ax=b has a solution for each b in ℝm. Your answer is correct. D. False. Since the columns of A span ℝm, the matrix A has a pivot position in exactly m−1 rows. f. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then A cannot have a pivot position in every row. Choose the correct answer below. A. False. Since the equation Ax=b has a solution for each b in ℝm, the equation Ax=b is consistent for each b in ℝm. B. False. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has a solution for each b in ℝm. C. True. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm. Your answer is correct. D. True. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the columns of A span ℝm.
a) False. The equation Ax=b is referred to as a matrix equation because A is a matrix. b) True. The equation Ax=b has the same solution set as the equation x1a1+x2a2+•••+x_na_n=b. c) False. If the augmented matrix [Ab] has a pivot position in every row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b. d) True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A. e) True. If the columns of A span ℝm, then the equation Ax=b has a solution for each b in ℝm. f) True. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm.
1.3.24 Mark each statement True or False. Justify each answer. Complete parts a through e below. a. When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line through v and the origin. A. False. Span{u,v} includes linear combinations of both u and v. B. False. Span{u,v} will not contain the origin. C. True. Span{u,v} is the set of all scalar multiples of u and all scalar multiples of v. b. Any list of five real numbers is a vector in ℝ5. A. False. A list of numbers is not enough to constitute a vector. B. False. A list of five real numbers is a vector in ℝn. C. False. A list of five real numbers is a vector in ℝ6. D. True. ℝ5 denotes the collection of all lists of five real numbers. c. Asking whether the linear system corresponding to an augmented matrix a1a2a3b has a solution amounts to asking whether b is in Span a1,a2,a3. A. False. If b corresponds to the origin then it cannot be in Span a1,a2,a3. B. True. An augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns. C. False. An augmented matrix having a solution does not mean b is in Span a1,a2,a3. d. The vector v results when a vector u−v is added to the vector v. A. False. Adding u−v to v results in u. B. True. Adding u−v to v results in v. C. False. Adding u−v to v results in 2v. D. False. Adding u−v to v results in u−2v. e. The weights c1,...,cp in a linear combination c1v1+...+cpvp cannot all be zero. A. True. Setting all the weights equal to zero results in the vector 0. B. True. Setting all the weights equal to zero does not result in the vector 0. C. False. Setting all the weights equal to zero does not result in the vector 0. D. False. Setting all the weights equal to zero results in the vector 0.
a) False. Span{u,v} includes linear combinations of both u and v. b) True. ℝ5 denotes the collection of all lists of five real numbers. c)True. An augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns. d) False. Adding u−v to v results in u. e) False. Setting all the weights equal to zero results in the vector 0.
3.1.39 Let A be an n×n matrix. Mark each statement as true or false. Justify each answer. a. An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices. b. The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column. a. Choose the correct answer below. A. The statement is true. The determinant of an n×n matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size (n−1)×(n−1). B. The statement is false. An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices only when n>3. Determinants of 1×1, 2×2, and 3×3 matrices are defined separately. C. The statement is true. The determinant of an n×n matrix A can be computed by a cofactor expansion along either diagonal. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size (n−1)×(n−1). D. The statement is false. Although determinants of (n−1)×(n−1) submatrices can be used to find n×n determinants, they are not involved in the definition of n×n determinants. b. Choose the correct answer below. A. The statement is false. The (i,j)-cofactor of A is the number Cij=detAij, where Aij is the submatrix obtained by deleting from A its ith row and jth column. B. The statement is false. The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its jth row and ith column. C. The statement is true. It is the definition of the (i,j)-cofactor of a matrix A. D. The statement is false. The (i,j)-cofactor of A is the number Cij=(−1)i+jdetAij, where Aij is the submatrix obtained by deleting from A its ith row and jth column.
a) The statement is true. The determinant of an n×n matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size (n−1)×(n−1). b) The statement is false. The (i,j)-cofactor of A is the number Cij=(−1)i+jdetAij, where Aij is the submatrix obtained by deleting from A its ith row and jth column.
2.2.10 Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. a. A product of invertible n×n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. A. False; if A and B are invertible matrices, then (AB)−1=B−1A−1. B. False; if A and B are invertible matrices, then (AB)−1=BA−1B−1. C. True; since invertible matrices commute, (AB)−1=B−1A−1=A−1B−1. D. True; if A and B are invertible matrices, then (AB)−1=A−1B−1. b. If A is invertible, then the inverse of A−1 is A itself. A. False; it does not follow from the fact that A is invertible that A−1 is also invertible. B. True; A is invertible if and only if A=A−1. Since A is invertible, A=A−1. Since A=A−1, A−1 is invertible, and A is the inverse of A−1. C. True; since A−1 is the inverse of A, A−1A=I=AA−1. Since A−1A=I=AA−1, A is the inverse of A−1. D. False; since inverses are not unique, it is possible that B≠A is the inverse of A−1. c. If A=abcd and ad=bc, then A is not invertible. A. True; A=abcd is invertible if and only if a≠b and b≠d. B. False; if A is invertible, then ad=bc. C. True; if ad=bc then ad−bc=0, and 1ad−bcd−b−ca is undefined. D. False; if ad=bc, then A is invertible. d. If A can be row reduced to the identity matrix, then A must be invertible. A. False; not every matrix that is row equivalent to the identity matrix is invertible. B. False; since the identity matrix is not invertible, A is not invertible either. C. True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible. Your answer is correct. D. True; since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible. e. If A is invertible, then elementary row operations that reduce A to the identity In also reduce A−1 to In. A. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E1E2E3•••Ep. Then the row operations required to reduce A−1 to the identity would correspond to the product E1−1E2−1E3−1•••Ep−1. B. True; by using the same row operations in reversed order A−1 may be reduced to the identity. C. True; if A is invertible then A is the product of some number of elementary matrices E1E2E3•••Ep, each corresponding to row operations. Then A−1 is Ep•••E3E2E1, the same elementary matrices. D. False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E1E2E3•••Ep. Then the row operations required to reduce A−1 to the identity would correspond to the product Ep−1•••E3−1E2−1E1−1.
a) False; if A and B are invertible matrices, then (AB)−1=B−1A−1. b) True; since A−1 is the inverse of A, A−1A=I=AA−1. Since A−1A=I=AA−1, A is the inverse of A−1. c) True; if ad=bc then ad−bc=0, and 1ad−bcd−b−ca is undefined. d) True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible. e) False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E1E2E3•••Ep. Then the row operations required to reduce A−1 to the identity would correspond to the product Ep−1•••E3−1E2−1E1−1.
1.4.32 Could a set of three vectors in ℝ^4 span all of ℝ^4? Explain. What about n vectors in ℝ^m when n is less than m? Could a set of three vectors in ℝ^4 span all of ℝ^4? Explain. Choose the correct answer below. A. No. There is no way for any number of vectors in ℝ^4 to span all of ℝ^4. B. Yes. Any number of vectors in ℝ^4 will span all of ℝ^4. C. Yes. A set of n vectors in ℝ^m can span ℝ^m when n<m. There is a sufficient number of rows in the matrix A formed by the vectors to have enough pivot points to show that the vectors span ℝ^m. D. No. The matrix A whose columns are the three vectors has four rows. To have a pivot in each row, A would have to have at least four columns (one for each pivot.) Could a set of n vectors in ℝ^m span all of ℝ^m when n is less than m? Explain. Choose the correct answer below. A. No. The matrix A whose columns are the n vectors has m rows. To have a pivot in each row, A would have to have at least m columns (one for each pivot.) B. No. Without knowing values of n and m, there is no way to determine if n vectors in ℝ^m will span all of ℝ^m. C. Yes. Any number of vectors in ℝ^m will span all of ℝ^m. D. Yes. A set of n vectors in ℝ^m can span ℝ^m if n<m. There is a sufficient number of rows in the matrix A formed by the vectors to have enough pivot points to show that the vectors span ℝ^m.
a) No. The matrix A whose columns are the three vectors has four rows. To have a pivot in each row, A would have to have at least four columns (one for each pivot.) b) No. The matrix A whose columns are the n vectors has m rows. To have a pivot in each row, A would have to have at least m columns (one for each pivot.)
3.1.22 Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. [8955] , [895+8k5+9k] What is the elementary row operation? A. Replace row 2 with k times row 2. B. Replace row 2 with k times row 1. C. Replace row 2 with k times row 1 plus row 2. D. Replace row 2 with row 1 plus k times row 2. How does the row operation affect the determinant? A. The determinant is decreased by 72k. B. The determinant is increased by 144k. C. The determinant is increased by 72k. D. The determinant does not change.
a) Replace row 2 with k times row 1 plus row 2. b) The determinant does not change.
3.1.20 Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. [abcd] , [a+kcb+kdcd] What is the elementary row operation? A. Row 1 is multiplied by k. B. Row 1 is replaced with the sum of itself and k times row 2. C. Row 2 is replaced with the sum of itself and k times row 1. D. Row 2 is multiplied by k. E. Rows 1 and 2 are interchanged. How does the row operation affect the determinant? A. It increases the determinant by k. B. It multiplies the determinant by k. C. It changes the sign of the determinant. D. It does not change the determinant.
a) Row 1 is replaced with the sum of itself and k times row 2. b) It does not change the determinant.
2.2.20 Suppose A, B, and X are n×n matrices with A, X, and A−AX invertible, and suppose that (A−AX)−1=X−1B. Complete parts (a) and (b) below. a. Explain why B is invertible. Choose the correct answer below. A. Multiply both sides of equation (A−AX)−1=X−1B by B−1 to obtain X−1=(A−AX)−1B−1. Now multiply both sides by (A−AX) to obtain the equation B−1=(A−AX)X−1. Thus, the inverse of B exists and B must be invertible. B. Solving the equation (A−AX)−1=X−1B for B yields X(A−AX)−1=B. Since X is invertible and (A−AX)−1 is invertible, the product X(A−AX)−1=B is also invertible. C. Since X−1B is equal to (A−AX)−1 and (A−AX)−1 is invertible, X−1B is also invertible. Since X−1B is invertible and X−1 is invertible, B must be invertible. D. Solving the equation (A−AX)−1=X−1B for B yields (A−AX)−1X=B. Since X is invertible and (A−AX)−1 is invertible, the product (A−AX)−1X=B is also invertible. b. Solve the equation (A−AX)−1=X−1B for X. X=_?_
a) Solving the equation (A−AX)−1=X−1B for B yields X(A−AX)−1=B. Since X is invertible and (A−AX)−1 is invertible, the product X(A−AX)−1=B is also invertible. b) (A+B^-1)^-1A
1.2.26 Suppose the coefficient matrix of a linear system of four equations in four variables has a pivot in each column. Explain why the system has a unique solution. What must be true of a linear system for it to have a unique solution? Select all that apply. A. The system has at least one free variable. B. The system has no free variables. Your answer is correct. C. The system is inconsistent. D. The system is consistent. Your answer is correct. E. The system has exactly one free variable. F. The system has one more equation than free variable. Use the given assumption that the coefficient matrix of the linear system of four equations in four variables has a pivot in each column to determine the dimensions of the coefficient matrix. The coefficient matrix has _?_ rows and _?_ columns. Choices: five, three, four five, three, four
a) The system has no free variables. The system is consistent. b) four, four
2.1.28 View vectors in ℝn as n×1 matrices. For u and v in ℝn, the matrix product uTv is a 1×1 matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product uvT is an n×n matrix, called the outer product of u and v. Suppose u and v are in ℝn. How are uTv and vTu related? How are uvT and vuT related? How are uTv and vTu related? A. The matrices uTv and vTu are related by uTv=vTuT, because uTv=uTvT=vTuTT=vTuT. B. The matrices uTv and vTu are related by uTv≠vTu, because uTv≠uTvT=vTuTT=vTu. C. The matrices uTv and vTu are related by uTv=−vTu, because uTv=uTvT=uTTvT=−vTu. D. The matrices uTv and vTu are related by uTv=vTu, because uTv=uTvT=vTuTT=vTu. How are uvT and vuT related? A. The matrices uvT and vuT are related by uvT=vuT, because uvT=vTTuT=vuT. B. The matrices uvT and vuT are related by uvTT=vuT and vuTT=uvT, because uvTT=vTTuT=vuT. Your answer is correct. C. The matrices uvT and vuT are related by uvTT=vuTT and vuTT=uvTT, because uvTT=vTTuT=vuTT. D. The matrices uvT and vuT are related by uvTT=uTvvuT and vuTT=vTuuvT, because uvTT=vTTuT=uTvvuT.
a) The matrices uTv and vTu are related by uTv=vTu, because uTv=(uTv)T=(vTuT)T=vTu. b) The matrices uvT and vuT are related by uvTT=vuT and vuTT=uvT, because uvTT=vTTuT=vuT.
3.1.19 State the row operation performed below and describe how it affects the determinant. [abcd] , [ab6c6d] What row operation was performed? A. The row operation subtracts 6 from row 2. B. The row operation adds 6 to row 2. C. The row operation scales row 2 by one-sixth. D. The row operation scales row 2 by 6. How does this affect the determinant? A. The determinant is multiplied by 6. B. The determinant is divided by 6. C. The determinant is 0. D. The determinant is unchanged.
a) The row operation scales row 2 by 6. b) The determinant is multiplied by 6.
1.2.21 In parts (a) through (e) below, mark the statement True or False. Justify each answer. (a) In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. Is this statement true or false? A. The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix. Your answer is correct. B. The statement is true. It is possible for there to be several different sequences of row operations that row reduces a matrix. C. The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique. D. The statement is false. For each matrix, there is only one sequence of row operations that row reduces it. (b) The row reduction algorithm applies only to augmented matrices for a linear system. Is this statement true or false? A. The statement is false. It is possible to create a linear system such that the row reduction algorithm does not apply to the corresponding augmented matrix. B. The statement is true. The row reduction algorithm is only useful when it is used to find the solution of a linear system. C. The statement is true. Every matrix with at least two columns can be interpreted as the augmented matrix of a linear system. D. The statement is false. The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. (c) A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. Is this statement true or false? A. The statement is false. Not every linear system has basic variables. B. The statement is true. It is the definition of a basic variable. Your answer is correct. C. The statement is true. If a linear system has both basic and free variables, then each basic variable can be expressed in terms of the free variables. D. The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free variable, not a basic variable. (d) Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or false? A. The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. Your answer is correct. B. The statement is true. Regardless of whether a linear system has free variables, the solution set of the system can be expressed using a parametric description. C. The statement is true. Solving a linear system is the same as finding the solution set of the system. The solution set of a linear system can always be expressed using a parametric description. D. The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has no more than one solution. (e) If one row in an echelon form of an augmented matrix is 00050, then the associated linear system is inconsistent. Is this statement true or false? A. The statement is true. The indicated row corresponds to the equation 5=0. This equation is a contradiction, so the linear system is inconsistent. B. The statement is false. The indicated row corresponds to the equation 5x4=0, which does not by itself make the system inconsistent. Your answer is correct. C. The statement is false. The indicated row corresponds to the equation 5x4=0, which means the system is consistent. D. The statement is true. The indicated row corresponds to the equation 5x4=0. This equation is not a contradiction, so the linear system is inconsistent.
a) The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix. b) The statement is false. The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. c) The statement is true. It is the definition of a basic variable. d) The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. e) The statement is false. The indicated row corresponds to the equation 5x4=0, which does not by itself make the system inconsistent.
1.2.22 In parts (a) through (e) below, mark the statement True or False. Justify each answer. a. The echelon form of a matrix is unique. Choose the correct answer below. A. The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique. B. The statement is true. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed. C. The statement is false. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations. D. The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique. b. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. Choose the correct answer below. A. The statement is true. The pivot positions in a matrix are determined completely by the positions of the leading entries of each row which are dependent on row interchanges. B. The statement is true. Every pivot position is determined by the positions of the leading entries of a matrix in reduced echelon form. C. The statement is false. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix. Your answer is correct. D. The statement is false. The pivot positions in a matrix depend on the location of the pivot column. c. Reducing a matrix to echelon form is called the forward phase of the row reduction process. Choose the correct answer below. A. The statement is false. Reducing a matrix to echelon form is called the backward phase and reducing a matrix to reduced echelon form is called the forward phase. B. The statement is false. The forward phase does not depend on whether a matrix is in echelon form or reduced echelon form. C. The statement is true. The forward phase occurs when a linear system has both basic and free variables, which can only be determined by reducing a matrix to echelon form. D. The statement is true. Reducing a matrix to echelon form is called the forward phase and reducing a matrix to reduced echelon form is called the backward phase. d. Whenever a system has free variables, the solution set contains many solutions. Choose the correct answer below. A. The statement is true. If there are infinitely many solutions, then there must be at least one free variable. B. The statement is true. If a solution has at least one free variable, there are infinitely many solutions. C. The statement is false. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty. Your answer is correct. D. The statement is false. A system with one free variable will have only one solution. e. A general solution of a system is an explicit description of all solutions of the system. Choose the correct answer below. A. The statement is true. After applying the row reduction algorithm and generating a general solution of a system, the rightmost column displays all of the particular solutions of that system. B. The statement is false. Each different choice of a free variable produces the same solution of the system. C. The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system. Your answer is correct. D. The statement is false. A general solution is the result of an inconsistent system, which has no particular solution.
a) The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique. b) The statement is false. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix. c) The statement is true. Reducing a matrix to echelon form is called the forward phase and reducing a matrix to reduced echelon form is called the backward phase. d) The statement is false. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty. e) The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system.
2.1.16 Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. a. If A and B are 3×3 matrices and B=b1b2b3, then AB=Ab1+Ab2+Ab3. Choose the correct answer below. A. The statement is true. By the definition of matrix multiplication, if A is an m×n matrix and B is an n×p matrix, then AB=Ab1b2...bp=Ab1+Ab2+...+Abp. B. The statement is false. The matrix Ab1+Ab2+Ab3 is the correct size matrix, but the plus signs should be minus signs. C. The statement is false. The matrix Ab1+Ab2+Ab3 is a 3×1 matrix, and AB must be a 3×3 matrix. The plus signs should be spaces between the 3 columns. D. The statement is true. By the definition of matrix multiplication, if A is an m×n matrix and B is an n×p matrix, then the resulting matrix AB is the sum of the columns of A using the weights from the corresponding columns of B. b. The second row of AB is the second row of A multiplied on the right by B. Choose the correct answer below. A. The statement is false. The second row of AB is the second row of A multiplied on the left by B. B. The statement is true. Every row and column of AB is the corresponding row and column of A multiplied on the right by B. C. The statement is false. Let columni(A) denote the ith column of matrix A. Then columni(AB)=columni(A)B. The same is not true for the rows of AB. D. The statement is true. Let rowi(A) denote the ith row of matrix A. Then rowi(AB)=rowi(A)B. Letting i=2 verifies this statement. c. (AB)C=(AC)B Choose the correct answer below. A. The statement is true. The associative law of multiplication for matrices states that (AB)C=(AC)B. B. The statement is true. The associative law of multiplication for matrices states that (AB)C=B(AC). C. The statement is false. The associative law of multiplication for matrices states that A(BC)=(AB)C. D. The statement is false. The associative law of multiplication for matrices states that (AB)C=B(AC). d. (AB)T=ATBT Choose the correct answer below. A. The statement is false. The transpose of the product of two matrices is the product of the transpose of the first matrix and the second matrix, or (AB)T=ATB. B. The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or (AB)T=BTAT. Your answer is correct. C. The statement is true. The transpose of the product of two matrices is the product of the transposes of the individual matrices in the same order, or (AB)T=ATBT. D. The statement is true. Matrix multiplication is not commutative so the products must remain in the same order e. The transpose of a sum of matrices equals the sum of their transposes. Choose the correct answer below. A. The statement is true. This is a generalized statement that follows from the theorem (A+B)T=AT+BT. B. The statement is false. For matrices A and B whose sizes are appropriate to sum, (A+B)T=AT−BT. C. The statement is true. This is a generalized statement that follows from the theorem (A+B)T=AT−BT. D. The statement is false. For matrices A and B whose sizes are appropriate to sum, (A+B)T=AT+BT.
a) The statement is false. The matrix Ab1+Ab2+Ab3 is a 3×1 matrix, and AB must be a 3×3 matrix. The plus signs should be spaces between the 3 columns. b) The statement is true. Let rowi(A) denote the ith row of matrix A. Then rowi(AB)=rowi(A)B. Letting i=2 verifies this statement. c) The statement is false. The associative law of multiplication for matrices states that A(BC)=(AB)C. d) The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or (AB)T=BTAT. e) The statement is true. This is a generalized statement that follows from the theorem (A+B)T=AT+BT.
1.1.23 Indicate whether the statements given in parts (a) through (d) are true or false and justify the answer. a. Is the statement "Every elementary row operation is reversible" true or false? Explain. A. True, because interchanging can be reversed by scaling, and scaling can be reversed by replacement. B. False, because only scaling and interchanging are reversible row operations. C. False, because only interchanging is a reversible row operation. D. True, because replacement, interchanging, and scaling are all reversible. b. Is the statement "A 5×6 matrix has six rows" true or false? Explain. A. True, because a 5×6 matrix has six columns and six rows. B. True, because a 5×6 matrix has five columns and six rows. C. False, because a 5×6 matrix has five rows and six columns. D. False, because a 5×6 matrix has five rows and five columns c. Is the statement "The solution set of a linear system involving variables x1, ..., xn is a list of numbers s1, ..., sn that makes each equation in the system a true statement when the values s1, ..., sn are substituted for x1, ..., xn, respectively" true or false? Explain. A. False, because the description applies to a single solution. The solution set consists of all possible solutions. B. False, because the list of numbers s1, ..., sn is the solution set for a linear system involving the variables x1, ..., xn−1. C. True, because the solution set of a linear system will have the same number of elements as the list of the variables in the system. D. True, because the list of variables x1, ..., xn and the list of numbers s1, ..., sn have a one-to-one correspondence. d. Is the statement "Two fundamental questions about a linear system involve existence and uniqueness" true or false? Explain. A. True, because two fundamental questions address whether the equations of the linear system exist in n-dimensional space and whether they can exist in more than one instance of n-dimensional space. B. False, because two fundamental questions address whether it is possible to solve the system with row operations or whether a computer is necessary. C. False, because two fundamental questions address the type of row operations that can be used on the system and whether the linear operations fundamentally change the system. D. True, because two fundamental questions address whether the solution exists and whether there is only one solution.
a) True, because replacement, interchanging, and scaling are all reversible. b) False, because a 5×6 matrix has five rows and six columns. c) False, because the description applies to a single solution. The solution set consists of all possible solutions. d)True, because two fundamental questions address whether the solution exists and whether there is only one solution.
2.2.9 Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. a. In order for a matrix B to be the inverse of A, both equations AB=I and BA=I must be true. A. True; since AB=BA, AB=I if and only if BA=I. B. True, by definition of invertible. C. False; if AB=I and BC=I, then A is one inverse of B and C is possibly another inverse of B. D. False; it's possible that the product AB is defined and equals I, yet the product BA is not defined. b. If A and B are n×n and invertible, then A−1B−1 is the inverse of AB. A. False; B−1A−1 is the inverse of AB. B. True; (AB)−1=(BA)−1=A−1B−1. C. False; 1AB is the inverse of AB. D. True; A−1B−1=B−1A−1=(AB)−1. c. If A=abcd and ab−cd≠0, then A is invertible. A. True; A−1=1ab−cdd−b−ca and this expression is always defined when ab−cd≠0. B. True; A=abcd is invertible if and only if a≠c and b≠d. C. False; if ad−bc≠0, then A is invertible. D. False; if A is invertible, then ab=cd. d. If A is an invertible n×n matrix, then the equation Ax=b is consistent for each b in ℝn. A. False; the matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying Ax=b is row equivalent to the identity matrix. B. True; since A is invertible, A−1b=x for all x in ℝn. Multiply both sides by A and the result is Ax=b. C. True; since A is invertible, A−1b exists for all b in ℝn. Define x=A−1b. Then Ax=b. Your answer is correct. D. False; the matrix A satisfies Ax=b if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible. e. Each elementary matrix is invertible. A. False; every matrix that is not invertible can be written as a product of elementary matrices. At least one of those elementary matrices is not invertible. B. True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix. C. False; it is possible to perform row operations on an n×n matrix that do not result in the identity matrix. Therefore, not every elementary matrix is invertible. D. True; since every invertible matrix is a product of elementary matrices, every elementary matrix must be invertible.
a) True, by definition of invertible. b) False; B−1A−1 is the inverse of AB c) False; if ad−bc≠0, then A is invertible. d) True; since A is invertible, A−1b exists for all b in ℝn. Define x=A−1b. Then Ax=b. e) True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix.
1.5.23 a. A homogeneous equation is always consistent. A. False. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one nontrivial solution. Thus a homogenous equation is always inconsistent. B. True. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one nontrivial solution. Thus a homogenous equation is always consistent. C. True. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always consistent. Your answer is correct. D. False. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always inconsistent. b. The equation Ax=0 gives an explicit description of its solution set. A. False. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set. Your answer is correct. B. True. The equation Ax=0 gives an explicit description of its solution set. Solving the equation amounts to finding an implicit description of its solution set. C. True. Since the equation is solved, Ax=0 gives an explicit description of the solution set. D. False. Since the equation is solved, Ax=0 gives an implicit description of its solution set. c. The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable. A. True. The homogeneous equation Ax=0 has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable. B. False. The homogeneous equation Ax=0 always has the trivial solution. Your answer is correct. C. True. The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable which implies that the equation has a nontrivial solution. D. False. The homogeneous equation Ax=0 never has the trivial solution. d. The equation x=p+tv describes a line through v parallel to p. A. False. The effect of adding p to v is to move v in a direction parallel to the plane through p and 0. So the equation x=p+tv describes a plane through p parallel to v. B. False. The effect of adding p to v is to move p in a direction parallel to the plane through v and 0. So the equation x=p+tv describes a plane through v parallel to p. C. False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation x=p+tv describes a line through p parallel to v. Your answer is correct. D. True. The effect of adding p to v is to move p in a direction parallel to the line through v and 0. So the equation x=p+tv describes a line through v parallel to p. e. The solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the equation Ax=0. A. False. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b, and there exists a vector p such that p is a solution. Your answer is correct. B. False. The solution set could be the trivial solution. The statement is only true when the equation Ax=b is inconsistent for some given b, and there exists a vector p such that p is a solution. C. False. The solution set could be empty. The statement is only true when the equation Ax=b is inconsistent for some given b, and there exists a vector p such that p is a solution. D. True. The equation Ax=b is always consistent and there always exists a vector p that is a solution.
a) True. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always consistent. b) False. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set. c) False. The homogeneous equation Ax=0 always has the trivial solution. d) False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation x=p+tv describes a line through p parallel to v. e) False. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b, and there exists a vector p such that p is a solution.
1.8.21 Determine whether each statement below is true or false. Justify each answer. a. A linear transformation is a special type of function. A. False. A linear transformation is not a function because it maps more than one vector x to the same vector T(x). B. False. A linear transformation is not a function because it maps one vector x to more than one vector T(x). C. True. A linear transformation is a function from ℝ to ℝ that assigns to each vector x in ℝ a vector T(x) in ℝ. D. True. A linear transformation is a function from ℝn to ℝm that assigns to each vector x in ℝn a vector T(x) in ℝm. b. If A is a 3×5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is ℝ3. A. True. The domain is ℝ3 because A has 3 columns, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝm. B. False. The domain is actually ℝ, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝ. C. False. The domain is actually ℝ5, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝn. D. True. The domain is ℝ3 because A has 3 rows, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝm. c. If A is an m×n matrix, then the range of the transformation x ↦ Ax is ℝm. A. False. The range of the transformation is ℝn because the domain of the transformation is ℝm. B. True. The range of the transformation is ℝm, because each vector in ℝm is a linear combination of the columns of A. C. True. The range of the transformation is ℝm, because each vector in ℝm is a linear combination of the rows of A. D. False. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form Ax. d. Every linear transformation is a matrix transformation. A. True. Every linear transformation T(x) can be expressed as a multiplication of a vector A by a matrix x such as Ax. B. False. A matrix transformation is a special linear transformation of the form x ↦ Ax where A is a matrix. C. False. A matrix transformation not a linear transformation because multiplication of a matrix A by a vector x is not linear. D. True. Every linear transformation T(x) can be expressed as a multiplication of a matrix A by a vector x such as Ax. e. A transformation T is linear if and only if Tc1v1+c2v2=c1Tv1+c2Tv2 for all v1 and v2 in the domain of T and for all scalars c1 and c2. A. False. A transformation T is linear if and only if T(0)=0. B. False. A transformation T is linear if and only if T(cu)=cT(u) for all scalars c and all u in the domain of T. C. False. A transformation T is linear if and only if T(u+v)=T(u)+T(v) for all u, v in the domain of T. D. True. This equation correctly summarizes the properties necessary for a transformation to be linear.
a) True. A linear transformation is a function from ℝn to ℝm that assigns to each vector x in ℝn a vector T(x) in ℝm. b) False. The domain is actually ℝ5, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝn. c) False. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form Ax d) False. A matrix transformation is a special linear transformation of the form x ↦ Ax where A is a matrix. e) True. This equation correctly summarizes the properties necessary for a transformation to be linear.
1.4.31 Let A be a 3×2 matrix. Explain why the equation Ax=b cannot be consistent for all b in ℝ3. Generalize your argument to the case of an arbitrary A with more rows than columns. Why is the equation Ax=b not consistent for all b in ℝ3? A. When written in reduced row echelon form, any 3×2 matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side. Your answer is correct. B. When written in reduced row echelon form, any 3×2 matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with infinitely many solutions. C. When written in reduced row echelon form, any 3×2 matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent. D. When written in reduced row echelon form, any 3×2 matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent. Let A be an m×n matrix, where m>n. Why is Ax=b not consistent for all b in ℝm? A. When written in reduced row echelon form, any m×n matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side. Your answer is correct. B. If A has more rows than columns, the number of pivots cannot be determined without knowing more information about A, and so A cannot be consistent. C. When written in reduced row echelon form, any m×n matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent. D. When written in reduced row echelon form, any m×n matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent. E. When written in reduced row echelon form, any m×n matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with infinitely many solutions.
a) When written in reduced row echelon form, any 3×2 matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side. b) When written in reduced row echelon form, any m×n matrix will have at least one row of all zeros. When solving Ax=b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
3.2.27 Mark each statement True or False. Justify each answer. Complete parts (a) through (d) below. a. A row replacement operation does not affect the determinant of a matrix. A. False. Changing any of the entries in the matrix changes the determinant. B. False. If a row is replaced by the sum of that row and k times another row, then the new determinant is k times the old determinant. C. True. If a multiple of one row of a matrix A is added to another to produce a matrix B, then det B equals det A. D. True. Row operations don't change the solutions of the matrix equation Ax=b. b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (−1)r, where r is the number of row interchanges made during row reduction from A to U. A. True. The determinant is the product of the entries on the diagonal and the pivots are all on the diagonal. B. False. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant. C. True. If A=2304, then det A=8. D. False. The determinant is the product of the number of pivots and (−1)r. c. If the columns of A are linearly dependent, then det A=0. A. False. The columns of I are linearly dependent, yet det I =1. B. False. If det A=0, then A is invertible. C. True. If the columns of A are linearly dependent, then one of the columns is equal to another. D. True. If the columns of A are linearly dependent, then A is not invertible. d. det(A+B)=det A+det B A. True. Determinants are linear transformations. B. False. det(A+B)=(det A)(det B) C. False. If A=1001 and B=−100−1, then det(A+B)=0 and det A+det B=2. D. True. If A=2010 and B=3050, then det(A+B)=0 and det A+det B=0.
a) True. If a multiple of one row of a matrix A is added to another to produce a matrix B, then det B equals det A. b) False. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant. c) True. If the columns of A are linearly dependent, then A is not invertible d)False. If A=1001 and B=−100−1, then det(A+B)=0 and det A+det B=2.
1.4.24 Determine whether each of statements a through f below are true or false. Justify each answer. a. Every matrix equation Ax=b corresponds to a vector equation with the same solution set. Choose the correct answer below. A. True. The matrix equation Ax=b is simply another notation for the vector equation x_1a_1+x_2a_2+•••+x_na_n=b, where a_1, ..., a_n are the columns of A. B. True. The matrix equation Ax=b is simply another notation for the vector equation x_1a_1+x_2a_2+•••+x_na_n=b, where a_1, ..., a_n are the rows of A. C. False. The matrix equation Ax=b only corresponds to an inconsistent system of vector equations. D. False. The matrix equation Ax=b does not correspond to a vector equation with the same solution set. b. If the equation Ax=b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below. A. False. b is only included in the set spanned by the columns of A if Ax=b is inconsistent. B. False. Ax=b is only consistent if the values of b are nonzero. C. True. The equation Ax=b has a solution set if and only if A has a pivot position in every row. D. True. The equation Ax=b has a nonempty solution set if and only if b is a linear combination of the columns of A. c. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. Choose the correct answer below. A. True. The matrix A is the matrix of coefficients of the system of vectors. B. True. Ax can be written as a linear combination of vectors because any two vectors can be combined by addition. C. False. A and x can only be written as a linear combination of vectors if and only if in Ax=b, b is nonzero. D. False. A and x cannot be written as a linear combination because the matrices do not have the same dimensions. d. If the coefficient matrix A has a pivot position in every row, then the equation Ax=b is inconsistent. Choose the correct answer below. A. False. If A has a pivot position in every row, the echelon form of the augmented matrix could not have a row such as [0 0 0 1], and Ax=b must be consistent. Your answer is correct. B. True. A pivot position in every row of a matrix indicates an inconsistent system of equations because the augmented column will always be zeros. C. False. If a coefficient matrix A has a pivot position in every row, then the equation Ax=b may or may not be consistent. D. True. If A has a pivot position in every row, then the augmented matrix must have a row of all zeros, indicating an inconsistent system of equations. e. The solution set of a linear system whose augmented matrix is [a1a2a3b] is the same as the solution set of Ax=b, if A= [a1a2a3]. Choose the correct answer below. A. False. The solution set of a linear system whose augmented matrix is [a1a2a3b] is the same as the solution set of Ax=b if and only if x has the same number of rows as A. B. True. The linear system whose augmented matrix is [a1a2a3b] will have the same solution set as Ax=b if and only if b is nonzero. C. True. If A is an m×n matrix with columns [a1a2•••an], and b is a vector in ℝm, the matrix equation Ax=b has the same solution set as the system of linear equations whose augmented matrix is [a1a2•••anb] . Your answer is correct. D. False. If A is an m×n matrix with columns [a1a2•••an], then b cannot be found in ℝm, and the system is inconsistent. f. If A is an m×n matrix whose columns do not span ℝm, then the equation Ax=b is consistent for every b in ℝm. Choose the correct answer below. A. True. If the columns of A do not span ℝm, b may or may not span ℝm. B. False. If the columns of A do not span ℝm, Ax=b cannot be consistent. C. True. If Ax=b is consistent, then the rows of A must span ℝm . D. False. If the columns of A do not span ℝm, then A does not have a pivot position in every row, and row reducing [Ab] could result in a row of the form [00•••0c], where c is a nonzero real number.
a) True. The matrix equation Ax=b is simply another notation for the vector equation x_1a_1+x_2a_2+•••+x_na_n=b, where a_1, ..., a_n are the columns of A. b) True. The equation Ax=b has a nonempty solution set if and only if b is a linear combination of the columns of A. c) True. The matrix A is the matrix of coefficients of the system of vectors. d) False. If A has a pivot position in every row, the echelon form of the augmented matrix could not have a row such as [0 0 0 1], and Ax=b must be consistent. e) True. If A is an m×n matrix with columns [a1a2•••an], and b is a vector in ℝm, the matrix equation Ax=b has the same solution set as the system of linear equations whose augmented matrix is [a1a2•••anb] . f) False. If the columns of A do not span ℝm, then A does not have a pivot position in every row, and row reducing [Ab] could result in a row of the form [00•••0c], where c is a nonzero real number.
1.7.22 In parts (a) to (d) below, mark the statement True or False. a. Two vectors are linearly dependent if and only if they lie on a line through the origin. Choose the correct answer below. A. True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin. B. False. Two vectors are linearly dependent if one of the vectors is a multiple of the other. The larger vector will be further from the origin than the smaller vector. C. False. If two vectors are linearly dependent then the graph of one will be orthogonal, or perpendicular, to the other. D. True. Linearly dependent vectors must always intersect at the origin. b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. Choose the correct answer below. A. False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector. B. False. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others. C. True. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation Ax=0. D. True. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector. c. If x and y are linearly independent, and if z is in Span{x, y}, then {x, y, z} is linearly dependent. Choose the correct answer below. A. True. Since z is in Span{x, y}, z is a linear combination of x and y. Since z is a linear combination of x and y, the set {x, y, z} is linearly dependent. B. False. Since z is in Span{x, y}, z cannot be written as a linear combination of x and y. The set {x, y, z} is linearly independent. C. True. Vector z is in Span{x, y} and x and y are linearly independent, so z is a scalar multiple of x or of y. Since z is a multiple of x or y, the set {x, y, z} is linearly dependent. D. False. Vector z is in Span{x, y} and x and y are linearly independent, so z must also be linearly independent of x and y. The set {x, y, z} is linearly independent. d. If a set in ℝn is linearly dependent, then the set contains more vectors than there are entries in each vector. Choose the correct answer below. A. True. For a set in ℝn to be linearly dependent, it must contain more than n vectors. B. False. If a set in ℝn is linearly dependent, then the set contains more entries in each vector than vectors. C. True. There exists a set in ℝn that is linearly dependent and contains more than n vectors. One example is a set in ℝ2 consisting of three vectors where one of the vectors is a scalar multiple of another. D. False. There exists a set in ℝn that is linearly dependent and contains n vectors. One example is a set in ℝ2 consisting of two vectors where one of the vectors is a scalar multiple of the other.
a) True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin. b) False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector. c) True. Since z is in Span{x, y}, z is a linear combination of x and y. Since z is a linear combination of x and y, the set {x, y, z} is linearly dependent. d) False. There exists a set in ℝn that is linearly dependent and contains n vectors. One example is a set in ℝ2 consisting of two vectors where one of the vectors is a scalar multiple of the other.
2.3.12 For this exercise assume that all matrices are n×n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts (a) through (e). Justify each answer. a. If there is an n×n matrix D such that AD=I, then there is also an n×n matrix C such that CA=I. Choose the correct answer below. A. False, because matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation AD=I. This implies that CA=I is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible. B. False; it does not follow from the Invertible Matrix Theorem that if AD=I, then CA=I. C. True; by the Invertible Matrix Theorem, if there is an n×n matrix D such that AD=I, then it must be true that there is also an n×n matrix C such that CA=I. D. True; by the Invertible Matrix Theorem if AD=I then A, D, or both is/are the identity matrix. Therefore, CA=I. b. If the columns of A are linearly independent, then the columns of A span ℝn. Choose the correct answer below. A. True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span ℝn. B. True; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ℝn. C. False; by the Invertible Matrix Theorem if the columns of A are linearly dependent, then the columns of A must span ℝn. D. False; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A do not span ℝn. c. If the equation Ax=b has at least one solution for each b in ℝn, then the solution is unique for each b. A. True, but only for x≠0; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then the equation Ax=0 does not only have the trivial solution. B. False; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then the linear transformation x ↦ Ax does not map ℝn onto ℝn. C. False; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then matrix A is not invertible. If A is not invertible, then according to the invertible matrix theorem the solution is not unique for each b. D. True; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. d. If the linear transformation (x) ↦ Ax maps ℝn into ℝn, then A has n pivot positions. A. False; according to the Invertible Matrix Theorem, (x) ↦ Ax maps ℝn into ℝn, then A has n+2 pivot positions. B. False; the linear transformation (x) ↦ Ax will always map ℝn into ℝn for any n×n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) ↦ Ax maps ℝn onto ℝn. C. True; according to the Invertible Matrix Theorem if (x) ↦ Ax maps ℝn into ℝn then A is invertible, and if a matrix is invertible it has n pivot positions. D. True; the linear transformation (x) ↦ Ax will always map ℝn into ℝn for any n×n matrix. Therefore, according to the Invertible Matrix Theorem A has n pivot positions. e. If there is a b in ℝn such that the equation Ax=b is inconsistent, then the transformation x ↦ Ax is not one-to-one. A. True; according to the Invertible Matrix Theorem if there is a b in ℝn such that the equation Ax=b is inconsistent, then matrix A is invertible. B. False; according to the Invertible Matrix Theorem if there is a b in ℝn such that the equation Ax=b is inconsistent, then the linear transformation x ↦ Ax maps ℝn onto ℝn. C. True; according to the Invertible Matrix Theorem if there is a b in ℝn such that the equation Ax=b is inconsistent, then equation Ax=b does not have at least one solution for each b in ℝn and this makes A not invertible. D. False; according to the Invertible Matrix Theorem if there is a b in ℝn such that the equation Ax=b is inconsistent, then equation Ax=b has at least one solution for each b in ℝn and this makes A invertible.
a) True; by the Invertible Matrix Theorem, if there is an n×n matrix D such that AD=I, then it must be true that there is also an n×n matrix C such that CA=I. b) True; by the Invertible Matrix Theorem if the columns of A are linearly independent, then the columns of A must span ℝn. c) True; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. d)False; the linear transformation (x) ↦ Ax will always map ℝn into ℝn for any n×n matrix. According to the Invertible Matrix Theorem A has n pivot positions only if (x) ↦ Ax maps ℝn onto ℝn. e) True; according to the Invertible Matrix Theorem if there is a b in ℝn such that the equation Ax=b is inconsistent, then equation Ax=b does not have at least one solution for each b in ℝn and this makes A not invertible.
1.8.22 Mark each statement True or False. Justify each answer. Complete parts a through e. a. The range of the transformation x ↦ Ax is the set of all linear combinations of the columns of A. A. True; each image T(x) is of the form Ax. Thus, the range is the set of all linear combinations of the columns of A. Your answer is correct. B. False; each image T(x) is not of the form Ax. Thus, the range is not the set of all linear combinations of the columns of A. C. False; each image T(x) is of the form Ax. Thus, the range is not the set of all linear combinations of the columns of A. D. True; each image T(x) is not of the form Ax. Thus, the range is not the set of all linear combinations of the columns of A. b. Every matrix transformation is a linear transformation. A. False; every matrix transformation has the properties T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all u and v, in the domain of T and all scalars c. Your answer is not correct. B. True; every matrix transformation has the properties T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all u and v, in the domain of T and all scalars c. This is the correct answer. C. True; every matrix transformation has the property T(u+v)=T(u)+T(v), but not all matrix transformations have the property T(cu)=cT(u) for all u and v, in the domain of T and all scalars c. D. False; not every matrix transformation has the properties T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all u and v, in the domain of T and all scalars c. c. If T: ℝn → ℝm is a linear transformation and if c is in ℝm, then a uniqueness question is "Is c in the range of T?" A. False; the question "is c in the range of T?" is the same as "does there exist an x whose image is c?" This is an existence question. This is the correct answer. B. True; the question "is c in the range of T?" is the same as"does there exist an x whose image is c?" This is a uniqueness question. C. False; the question "is c in the range of T?" is the same as "is c the image of a unique x in ℝn?" This is an existence question. D. True; the question "is c in the range of T?" is the same as "is c the image of a unique x in ℝn?" This is a uniqueness question. Your answer is not correct. d. A linear transformation preserves the operations of vector addition and scalar multiplication. A. True; The linear transformation T(cu+dv) is the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are preserved. Your answer is correct. B. False; The linear transformation T(cu+dv) is the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are not preserved. C. True; The linear transformation T(cu+dv) is not the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are preserved. D. False; The linear transformation T(cu+dv) is not the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are not preserved. e. A linear transformation T: ℝn → ℝm always maps the origin of ℝn to the origin of ℝm. A. True; for a linear transformation, T(0) does not equal 0. B. False; for a linear transformation, T(0) does not equal 0. C. True; for a linear transformation, T(0) is equal to 0. Your answer is correct. D. False; for a linear transformation, T(0) is equal to 0.
a) True; each image T(x) is of the form Ax. Thus, the range is the set of all linear combinations of the columns of A. b) True; every matrix transformation has the properties T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all u and v, in the domain of T and all scalars c. c) False; the question "is c in the range of T?" is the same as "does there exist an x whose image is c?" This is an existence question. d) True; The linear transformation T(cu+dv) is the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are preserved. e) True; for a linear transformation, T(0) is equal to 0.
1.2.24 Suppose a system of linear equations has a 3×5 augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not? To determine if the linear system is consistent, use the portion of the Existence and Uniqueness Theorem, shown below. A linear system is consistent if and only if the rightmost column of the augmented matrix _?_ a pivot column. That is, if and only if an echelon form of the augmented matrix has _?_ [0 ... 0 b] with b nonzero. Choices : is not, is at least one row, one row In the augmented matrix described above, is the rightmost column a pivot column? Yes No In the echelon form of the augmented matrix, is there a row of the form [0 0 0 0 b] with b nonzero? No Yes Therefore, by the Existence and Uniqueness Theorem, the linear system is _?_ Choices: inconsistent consistent
a) is not b) no row c) no d) no e) consistent
1.7.30 (a) Fill in the blank in the following statement. If A is an m×n matrix, then the columns of A are linearly independent if and only if A has _______ pivot columns. (b) Explain why the statement in (a) is true. (a) Fill in the blank. If A is an m×n matrix, then the columns of A are linearly independent if and only if A has _?_ pivot columns. Choices: m,n,1,0 (b) Why is the statement in (a) true? A. The columns of a matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution. This happens if and only if Ax=0 has no free variables, meaning every variable is a basic variable, that is, if and only if there are 0 pivot columns. B. The columns of a matrix A are linearly independent if and only if Ax=0 has no free variables, meaning every variable is a basic variable, that is, if and only if every column of A is a pivot column. C. The columns of a matrix A are linearly independent if and only if the equation Ax=0 has more unknowns than equations so there must be free variables. This happens if and only if there are m pivot columns.
a) n b) The columns of a matrix A are linearly independent if and only if Ax=0 has no free variables, meaning every variable is a basic variable, that is, if and only if every column of A is a pivot column.
1.1.24 Indicate whether the statements given in parts (a) through (d) are true or false and justify the answer. a. Is the statement "Two matrices are row equivalent if they have the same number of rows" true or false? Explain. A. True, because two matrices that are row equivalent have the same number of solutions, which means that they have the same number of rows. B. True, because two matrices are row equivalent if they have the same number of rows and column equivalent if they have the same number of columns. C. False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other. Your answer is correct. D. False, because if two matrices are row equivalent it means that they have the same number of row solutions. b. Is the statement "Elementary row operations on an augmented matrix never change the solution set of the associated linear system" true or false? Explain. A. True, because the elementary row operations replace a system with an equivalent system. Your answer is correct. B. False, because the elementary row operations augment the number of rows and columns of a matrix. C. False, because the elementary row operations make a system inconsistent. D. True, because elementary row operations are always applied to an augmented matrix after the solution has been found. c. Is the statement "Two equivalent linear systems can have different solution sets" true or false? Explain. A. False, because two systems are called equivalent only if they both have no solution. B. True, because equivalent linear systems are systems that have the same number of rows and columns when they are written as augmented matrices, which means that they can have different solution sets. C. False, because two systems are called equivalent if they have the same solution set. Your answer is correct. D. True, because equivalent linear systems are systems with the same number of variables, which means that they can have different solution sets. d. Is the statement "A consistent system of linear equations has one or more solutions" true or false? Explain. A. False, because a consistent system has infinitely many solutions. B. True, because a consistent system is made up of equations for planes in three-dimensional space. C. False, because a consistent system has only one unique solution. D. True, a consistent system is defined as a system that has at least one solution.
a) False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other. b) True, because the elementary row operations replace a system with an equivalent system. c) False, because two systems are called equivalent if they have the same solution set. d) True, a consistent system is defined as a system that has at least one solution.
2.3.11 For this exercise assume that the matrices are all n×n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but "statement 1" is true. Complete parts a through e. Justify each answer. a. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n×n identity matrix. A. True; by the Invertible Matrix Theorem if equation Ax=0 has only the trivial solution, then the equation Ax=b has no solutions for each b in ℝn. Thus, A must also be row equivalent to the n×n identity matrix. B. False; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the n×n identity matrix. C. True; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n×n identity matrix. D. False; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span ℝn. Thus, A must also be row equivalent to the n×n identity matrix. b. If the columns of A span ℝn, then the columns are linearly independent. A. True; the Invertible Matrix Theorem states that if the linear transformation x ↦ Ax does not map ℝn into ℝn, then A is invertible. Therefore, the columns are linearly independent. B. False; the Invertible Matrix Theorem states that if the linear transformation x ↦ Ax maps ℝn into ℝn, then A is not invertible. Therefore, the columns are linearly dependent. C. True; the Invertible Matrix Theorem states that if the columns of A span ℝn, then matrix A is invertible. Therefore, the columns are linearly independent. D. False; the Invertible Matrix Theorem states that if the columns of A span ℝn, then matrix A is not invertible. Therefore, the columns are linearly dependent. c. If A is an n×n matrix, then the equation Ax=b has at least one solution for each b in ℝn. A. True; by the Invertible Matrix Theorem Ax=b has at least one solution for each b in ℝn for all matrices of size n×n. B. False; by the Invertible Matrix Theorem Ax=b has at least one solution for each b in ℝn only if a matrix is invertible. C. False; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then the equation Ax=b has no solution. D. True; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in ℝn, then the matrix is not invertible. d. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions. A. False; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then matrix A is invertible. Therefore, A has n pivot positions. B. False; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then the columns of A do not form a linearly independent set. Therefore, A has n pivot positions. C. True; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions. D. True; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then the columns of A form a linearly independent set. Therefore, A has fewer than n pivot positions. e. If AT is not invertible, then A is not invertible. A. False; by the Invertible Matrix Theorem if AT is not invertible all statements in the theorem are true, including A is invertible. Therefore, A is invertible. B. False; by the Invertible Matrix Theorem if AT is not invertible then there is not an n×n matrix C such that CA=I. Therefore, A is invertible. C. True; by the Invertible Matrix Theorem if AT is not invertible then there is an n×n matrix C such that CA=I. This means that A must not be invertible. D. True; by the Invertible Matrix Theorem if AT is not invertible all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.
a) True; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n×n identity matrix. b) True; the Invertible Matrix Theorem states that if the columns of A span ℝn, then matrix A is invertible. Therefore, the columns are linearly independent. c) False; by the Invertible Matrix Theorem Ax=b has at least one solution for each b in ℝn only if a matrix is invertible. d) True; by the Invertible Matrix Theorem if the equation Ax=0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions. e) True; by the Invertible Matrix Theorem if AT is not invertible all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.
1.9.23 In parts (a) through (e) below, mark the statement True or False. Justify each answer. a. A linear transformation T: ℝn→ℝm is completely determined by its effect on the columns of the n×n identity matrix. Choose the correct answer below. A. True. The vector x can be written as a linear combination of the columns of the identity matrix. T is a linear transformation so T(x) can be written as a linear combination of the vectors T(e1) and T(e2). Your answer is correct. B. True. A transformation T is defined in terms of its effect on the columns of the n×n identity matrix because otherwise the standard matrix A would be impossible to determine. C. False. It is possible for more than one linear transformation to have the same effect on the columns of the n×n identity matrix. D. False. The transformation is jointly determined by its effect on the columns of the n×n identity matrix and by the result of writing x as a linear combination of the columns of the standard matrix A. b. If T: ℝ2→ℝ2 rotates vectors about the origin through an angle φ, then T is a linear transformation. Choose the correct answer below. A. True. The standard matrix, A, of the linear transformation is sinφcosφ−cosφsinφ. B. False. Rotations move points around a circle and so are not linear transformations. C. True. The standard matrix, A, of the linear transformation is cosφ−sinφsinφcosφ. Your answer is correct. D. False. The standard matrix, A, of the linear transformation is cosφsinφ−sinφ−cosφ, but because the sine and cosine functions are nonlinear, the transformation is not linear. c. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. Choose the correct answer below. A. True. When one transformation is applied after another, the property of a linear transformation which reads T(u+v)=T(u)+T(v) for vectors u and v, will not be true. In these cases, T(u+v) instead equals T(u)T(v). B. False. A transformation is linear if T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all vectors u, v, and scalars c. The first transformation results in some vector u, so the properties of a linear transformation must still apply when two transformations are applied. Your answer is correct. C. False. The combined effect of two linear transformations is always linear because multiplying two linear functions together will result in a function which is also linear. D. True. When different types of transformations are combined, such as a rotation and a skew, the transformation is not linear except for a few special cases. d. A mapping T: ℝn→ℝm is onto if every vector x in ℝn maps onto some vector in ℝm. Choose the correct answer below. A. True. A transformation is onto when all possible values in the codomain, or range, are mapped to by some value in the domain. B. False. A mapping T: ℝn→ℝm is onto if every vector in ℝm is mapped onto by some vector x in ℝn. Your answer is correct. C. False. A mapping T: ℝn→ℝm is one-to-one if every vector x in ℝn maps onto some vector in ℝm. D. True. A transformation is onto when, for every b in the codomain, the matrix equation Ax=b has a unique solution. e. If A is a 3×2 matrix, then the transformation x↦Ax cannot be one-to-one. Choose the correct answer below. A. False. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If Ax=b does not have a free variable, then the transformation represented by A is one-to-one. Your answer is correct. B. False. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be 1×4 and still represent a one-to-one transformation. C. True. A transformation is one-to-one only if the columns of A are linearly independent and a 3×2 matrix cannot have linearly independent columns. D. True. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column.
a)True. The vector x can be written as a linear combination of the columns of the identity matrix. T is a linear transformation so T(x) can be written as a linear combination of the vectors T(e1) and T(e2). b) True. The standard matrix, A, of the linear transformation is cosφ−sinφsinφcosφ. c) False. A transformation is linear if T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all vectors u, v, and scalars c. The first transformation results in some vector u, so the properties of a linear transformation must still apply when two transformations are applied. d) False. A mapping T: ℝn→ℝm is onto if every vector in ℝm is mapped onto by some vector x in ℝn. e) False. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If Ax=b does not have a free variable, then the transformation represented by A is one-to-one.
1.5.30 Let A be a 3×3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. a. Does the equation Ax=0 have a nontrivial solution? A. Yes. Since A has 2 pivots, there is one free variable. So Ax=0 has a nontrivial solution. B. No. Since A has 2 pivots, there are no free variables. With no free variables, Ax=0 has only the trivial solution. C. Yes. Since A has 2 pivots, there is one free variable. The solution set of Ax=0 does not contain the trivial solution if there is at least one free variable. D. No. Since A has 2 pivots, there is one free variable. Since there is at least one free variable, Ax=0 has only the trivial solution. b. Does the equation Ax=b have at least one solution for every possible b? A. Yes. Since A has 2 pivots, there are no free variables. So there is at least one solution for every possible b. B. Yes. A has a free variable. So the free variable can equal any value such that there is at least one solution for every possible b. C. No. A has one free variable. To have at least one solution for every possible b, A cannot have any free variable. D. No. A has one free variable, so there will be no solution to the system for any possible b.
a)Yes. Since A has 2 pivots, there is one free variable. So Ax=0 has a nontrivial solution. b) No. A has one free variable. To have at least one solution for every possible b, A cannot have any free variable
1.2.29 A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain. Choose the correct answer below. A. No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution. B. Yes, it can have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains either a unique solution or infinitely many solutions. If the linear system is inconsistent, there is no solution. C. No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If there is a free variable, the solution set contains a unique solution. D. Yes, it can have a unique solution. Because there are more equations than variables, there are no free variables. If the system is consistent and there are no free variables, the solution set contains a unique solution. If the system is inconsistent, there is no solution
No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution.