linear algebra
Give a formula for (ABx)^T , where x is a vector and A and B are matrices of appropriate size.
(ABx)^T = x^T*B^T*A^T, because (ABx)^T = x^T(AB)^T = x^T*B^T*A^T
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer.
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the nxn matrix has n pivot positions.
If the n×n matrices E and F have the property that EF=I, then E and F commute. Explain why.
According the Invertible Matrix Theorem, E and F must be invertible and inverses. So FE=I and I=EF. Thus, E and F commute
Solve the equation AB=BC for A, assuming that A, B, and C are square matrices and B is invertible. A = ________
BCB^-1
If the columns of a 7×7 matrix D are linearly independent, what can you say about the solutions of Dx=b? Why?
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.
ℝ2 is a subspace of ℝ3.Is this statement true or false?
False because ℝ2 is not even a subset of R3
det(A+B)=det A+det B
False. If A=[1,0,0,1] and B = [-1,0,0,-1] , then det(A+B)=0 and det A+det B=2.
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.
False. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant.
If A is an n×n matrix, then the equation Ax=b has at least one solution for each b in ℝn.
False; by the Invertible Matrix Theorem Ax=b has at least one solution for each b in ℝn only if a matrix is invertible.
If A = [ a,b,c,d] and ab-cd ≠ 0, then A is invertible.
False; if ad-bc ≠ 0, then A is invertible
Let H be the set of all points in the xy-plane having at least one nonzero coordinate, that is, H=xy : x, y not both zero. Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy: (not both zero)
H is not a vector space because it fails to satisfy all three properties.
Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy.
H is not a vector space because it is not closed under multiplication by scalars.
State which property of determinants is illustrated in this equation.
If a multiple of one row of A is added to another row to produce matrix B, then det B=det A.
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?]
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.
Let r1, ..., rp be vectors in ℝn, and let Q be an M×n matrix. Write the matrix Qr1⋯Qrp as a product of two matrices (neither of which is an identity matrix). If the matrix R is defined asr1⋯rp,then the matrix Qr1⋯Qrp can be written as ________
If the matrix R is defined as r1⋯rp , then the matrix Qr1⋯Qrp can be written as QR
State which property of determinants is illustrated in this equation.
If two rows of A are interchanged to produce B, then det B = -det A
Choose the correct theorem that indicates why these vectors show that W is a subspace of ℝ3.
If v1,...,vp are in a vector space V, then Span{v1,...,vp} is a subspace of V.
If A is invertible, then the columns of A^−1 are linearly independent. Explain why.
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.
If C is 6×6 and the equation Cx=v is consistent for every v inℝ6, is it possible that for some v, the equation Cx=v has more than one solution? Why or why not?
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.
Is it possible for a 5×5 matrix to be invertible when its columns do not span ℝ5? Why or why not?
It is not possible; according to the Invertible Matrix Theorem an n×n matrix cannot be invertible when its columns do not span ℝn.
How can the third column ofA^−1 be found without computing the other columns?
Row reduce the augmented matrix [Ae3], where e3 is the third column of I3.
Why does this show that H is a subspace of ℝ3?
Since v is in ℝ3, H=Span{v} is a subspace of ℝ3.
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.
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.
Let U be the 32 matrix . The first column of U lists the costs per dollar of output for manufacturing product B, and the second column lists the costs per dollar of output for manufacturing product C. The first row is the cost of materials, the second row is the cost of labor, and the third row is the cost of overhead. Let be a vector in that lists the output (measured in dollars) of products B and C manufactured during the first quarter of the year, and let , , and be the analogous vectors that list the amounts of products B and C manufactured in the second, third, and fourth quarters, respectively. Give an economic desciption of the data in the matrix UQ, where .
The 4 columns of UQ list the total costs for materials, labor, and overhead used to manufacture products B and C during the 4 quarters of the year.
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?
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
If an n×n matrix K cannot be row reduced to In, what can you say about the columns of K? Why?
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.
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?
The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
The matrix is invertible because its determinant is not zero.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.(4x4)
The matrix is invertible. The given matrix has 4 pivot positions.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.(3x3 0 top right triangle)
The matrix is invertible. The given matrix has three pivot positions.
Can a square matrix with two identical columns be invertible? Why or why not?
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.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.(3x3 0 diagonal)
The matrix is not invertible. If the given matrix is A, A is not row equivalent to the nxn identity matrix.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.(3x3 0 middle column)
The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set.
How many vectors are in {v1,v2,v3}? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The number of vectors in Span{v1,v2,v3} is enter your response here 3
The set M2×2 of all 2×2 matrices is a vector space, under the usual operations of addition of matrices and multiplication by real scalars. Determine if the set H of all matrices of the form [a,b,0,d] is a subspace ofM2×2.
The set H is a subspace of M2×2 because H contains the zero vector of M2×2, H is closed under vector addition, and H is closed under multiplication by scalars.
Determine if the given set is a subspace of ℙ8. Justify your answer. The set of all polynomials of the form p(t) =at8, where a is in ℝ.
The set is a subspace of ℙ8. The set contains the zero vector of ℙ8, the set is closed under vector addition, and the set is closed under multiplication by scalars.
Is the set of vectors linearly independent?
The set of vectors is linearly independent, because the determinant is not zero.
b. Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A. Choose the correct answer below
The statement is false. The definition of AB states that each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
The transpose of a product of matrices equals the product of their transposes in the same order. Choose the correct answer below.
The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order.
The (i,j)-cofactor of a matrix A is the matrixAij obtained by deleting from A its ith row and jth column.
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.
An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices.
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).
c. AB+AC=A(B+C)
The statement is true. The distributive law for matrices states that A(B+C)=AB+AC.
A^T +B^T = (A+B)^T
The statement is true. The transpose property states that (A+B)^T = A^T +B^T
How many vectors are in span{v1,v2,v3}? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
There are infinitely many vectors in Span{v1,v2,v3}.
If u is a vector in a vector space V, then (−1)u is the same as the negative of u. Is this statement true or false?
True because for each u in V, −u=(−1)u
A vector space is also a subspace of itself. Is this statement true or false?
True because the axioms for a vector space include all the conditions for being a subspace
A vector is any element of a vector space.Is this statement true or false?
True by the definition of a vector space
A row replacement operation does not affect the determinant of a matrix.
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.
If the columns of A are linearly dependent, then det A=0.
True. If the columns of A are linearly dependent, then A is not invertible.
If A^T is not invertible, then A is not invertible.
True; by the Invertible Matrix Theorem if A^T is not invertible all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
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.
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.
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.
If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in R^n.
True; since A is invertible,A^-1*b exists for all b in R^n. Define x=A^-1*b. Then Ax=b.
If the columns of A span ℝn, then the columns are linearly independent.
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.
Is w in the subspace spanned by {v1,v2,v3}?
Vector w is in the subspace spanned by {v1,v2,v3} because w is a linear combination of v1,v2, and v3, which can be seen because any echelon form of the augmented matrix of the system has no row of the form [0••• 0 b] with b≠0.
is w in {v1,v2,v3}? How many vectors are in {v1,v2, v3}?
Vector w is not in {v1,v2,v3} because it is not v1,v2, or v3.
. If A and B are 2×2 with columns a1,a2, and b1,b2, respectively, then AB=a1b1a2b2. Fill in the blanks. The statement is ________________ The definition of matrix multiplication states that if A is an m×n matrix and B is an n×p matrix with columns b1,...,bp, then AB=Upper A left bracket Start 1 By 4 Matrix 1st Row 1st Column Bold b 1 2nd Column Bold b 2 3rd Column ... 4st Column Bold b Subscript p EndMatrix right bracket equals left bracket Start 1 By 4 Matrix 1st Row 1st Column Upper A Bold b 1 2nd Column Upper A Bold b 2 3rd Column ... 4st Column Upper A Bold b Subscript p EndMatrix right bracket .
false, AB = A[b1 b2 bp] =Ab1 Ab2 Abp
If A and B are nxn and invertible, then A^-1*B^-1 is the inverse of AB.
False; B^-1*A ^-1 is the inverse of AB.
A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u +v are in H, and (iii) c is a scalar and cu is in H. Is this statement true or false?
False; parts (ii) and (iii) should state that u and v represent all possible elements of H.
In order for a matrix B to be the inverse of A, both equations AB= I and BA= I must be true.
True, by definition of invertible.
Each elementary matrix is invertible.
True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix.