Linear Algebra Final Review T/F

a. Another notation for the vector −43 is −43. Choose the correct answer below. A. False. The alternative notation for a​ (column) vector is ​(−4​,3​), using parentheses and commas. B. False. The matrices are not equal because they have different​ entries, even though they have the same shape. C. True. The matrices both represent the geometric point ​(−4​,3​). D. True. The matrices are equal because they have the same​ entries, even though they have different shapes.

A. False. The alternative notation for a​ (column) vector is ​(−4​,3​), using parentheses and commas.

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. B. False. Since the equation is​ solved, Ax=0 gives an implicit description of its solution set. C. 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. D. True. Since the equation is​ solved, Ax=0 gives an explicit description of the 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.

Could a set of three vectors in ℝ4 span all of ℝ4​? Explain. Choose the correct answer below. 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. There is no way for any number of vectors in ℝ4 to span all of ℝ4. C. Yes. Any number of vectors in ℝ4 will span all of ℝ4. D. Yes. A set of n vectors in ℝm can span ℝm 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 ℝm.

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).

c. The orthogonal projection y of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute y. Choose the correct answer below. A. The statement is false because the uniqueness property of the orthogonal decomposition y=y+z indicates​ that, no matter the basis used to find​ it, it will always be the same. B. The statement is false because the orthogonal projection y of y onto a subspace W depends on an orthonormal basis for W. C. The statement is true because the orthogonal projection y of y onto a subspace W depends on an orthonormal basis for W. D. The statement is true because for each different orthogonal​ basis, y is expressed as a different linear combination of the vectors in that basis.

A. The statement is false because the uniqueness property of the orthogonal decomposition y=y+z indicates​ that, no matter the basis used to find​ it, it will always be the same.

b. If A is​ diagonalizable, then A has n distinct eigenvalues. A. The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors. B. The statement is true. A diagonalizable matrix must have exactly n eigenvalues. C. The statement is true. A diagonalizable matrix must have n distinct eigenvalues. D. The statement is false. A diagonalizable matrix must have more than n eigenvalues.

A. The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.

d. If A is​ invertible, then A is diagonalizable. A. The statement is false. An invertible matrix may have fewer than n linearly independent​ eigenvectors, making it not diagonalizable. B. The statement is true. A diagonalizable matrix is​ invertible, so an invertible matrix is diagonalizable. C. The statement is false. Invertible matrices always have a maximum of n linearly independent​ eigenvectors, making it not diagonalizable. D. The statement is true. If a matrix is​ invertible, then it has n linearly independent ​eigenvectors, making it diagonalizable.

A. The statement is false. An invertible matrix may have fewer than n linearly independent​ eigenvectors, making it not diagonalizable.

​(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. B. 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. C. The statement is true. It is possible for there to be several different sequences of row operations that row reduces a matrix. D. The statement is false. For each​ matrix, there is only one sequence of row operations that row reduces it.

A. The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.

b. For each y and each subspace​ W, the vector y−projWy is orthogonal to W. Choose the correct answer below. A. The statement is true because y can be written uniquely in the form y=projWy+z where projWy is in W and z is in W⊥ and it follows that z=y−projWy. B. The statement is false because y can be written uniquely in the form y=projWy+z where z is in W and projWy is in W⊥ and it follows that z=y−projWy. C. The statement is false because y−projWy is in W and so cannot be orthogonal to W. D. The statement is true because y and projWy are both orthogonal to W.​ Thus, a linear combination of them must also be orthogonal to W.

A. The statement is true because y can be written uniquely in the form y=projWy+z where projWy is in W and z is in W⊥ and it follows that z=y−projWy.

d. The column space of a matrix A is the set of solutions of Ax=b. A. This statement is false. The column space of A is the set of all b for which Ax=b has a solution. B. This statement is false. The column space of A is the set of all A for which Ax=b has a solution. C. This statement is false. The column space of a matrix A is the set of solutions of Ax=0. D. This statement is true. This is the definition of a column space.

A. This statement is false. The column space of A is the set of all b for which Ax=b has a solution.

c. The null space of an m×n matrix is a subspace of ℝn. A. This statement is true. For an m×n matrix​ A, the solutions of Ax=0 are vectors in ℝn and satisfy the properties of a vector space. B. This statement is false. The null space of a matrix does not contain the zero vector. C. This statement is false. For an m×n matrix​ A, the solutions of Ax=0 belong to ℝm. D. This statement is false. This set is not closed under scalar multiplication.

A. This statement is true. For an m×n matrix​ A, the solutions of Ax=0 are vectors in ℝn and satisfy the properties of a vector space.

c. The columns of any 4×5 matrix are linearly dependent. Choose the correct answer below. A. 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. 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. 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.

A. 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.

a. A homogeneous equation is always consistent. 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. 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. C. 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. D. 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.

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.

a. Not every linearly independent set in ℝn is an orthogonal set. A. True. For​ example, the vectors 01 and 11 are linearly independent but not orthogonal. B. False. For​ example, in every linearly independent set of two vectors in ℝ2​, one vector is a multiple of the​ other, so the vectors cannot be orthogonal. C. True. For​ example, the vectors 1−1 and 11 are linearly independent but not orthogonal. D. False. Every orthogonal set is linearly independent.

A. True. For​ example, the vectors 01 and 11 are linearly independent but not orthogonal.

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. 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. B. 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. C. 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. D. 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.

A. 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.

d. The solution set of the linear system whose augmented matrix is a1a2a3b is the same as the solution set of the equation x1a1+x2a2+x3a3=b. Choose the correct answer below. A. True. The augmented matrix for x1a1+x2a2+x3a3=b is a1a2a3b. B. True. The augmented matrix for x1a1+x2a2+x3a3=b is x1x2x3b. C. False. The augmented matrix for x1a1+x2a2+x3a3=b is not a1a2a3b. D. False. The solution set for an augmented matrix is not the same as the solution set of an equation.

A. True. The augmented matrix for x1a1+x2a2+x3a3=b is a1a2a3b.

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. ​False; 1AB is the inverse of AB. C. ​True; A−1B−1=B−1A−1=(AB)−1. D. ​True; (AB)−1=(BA)−1=A−1B−1.

A. ​False; B−1A−1 is the inverse of AB.

e. Each elementary matrix is invertible. A. ​True; since each elementary matrix corresponds to a row​ operation, and every row operation is​ reversible, every elementary matrix has an inverse matrix. B. ​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. C. ​True; since every invertible matrix is a product of elementary​ matrices, every elementary matrix must be invertible. D. ​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.

A. ​True; since each elementary matrix corresponds to a row​ operation, and every row operation is​ reversible, every elementary matrix has an inverse matrix.

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. 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. 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. 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 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.

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.

b. The points in the plane corresponding to −25 and −52 lie on a line through the origin. Choose the correct answer below. A. False. −25 and −52 correspond to arrows that meet at right angles. B. False. If −25 and −52 were on the line through the​ origin, then −52 would be a multiple of −25​, which is not the case. C. True. −25 and −52 correspond to arrows pointing in precisely opposite directions and so are on the same line through the origin. D. True. −25 and −52 are on a line through the origin since −52 is a positive multiple of −25.

B. False. If −25 and −52 were on the line through the​ origin, then −52 would be a multiple of −25​, which is not the case.

c. The orthogonal projection y of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute y. Choose the correct answer below. A. The statement is true because for each different orthogonal​ basis, y is expressed as a different linear combination of the vectors in that basis. B. The statement is false because the uniqueness property of the orthogonal decomposition y=y+z indicates​ that, no matter the basis used to find​ it, it will always be the same. C. The statement is true because the orthogonal projection y of y onto a subspace W depends on an orthonormal basis for W. D. The statement is false because the orthogonal projection y of y onto a subspace W depends on an orthonormal basis for W.

B. The statement is false because the uniqueness property of the orthogonal decomposition y=y+z indicates​ that, no matter the basis used to find​ it, it will always be the same.

a. A matrix A is diagonalizable if A has n eigenvectors. A. The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors. B. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors. C. The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector. D. The statement is false. A matrix is diagonalizable if and only if it has n−1 linearly independent eigenvectors.

B. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.

(b) The row reduction algorithm applies only to augmented matrices for a linear system. Is this statement true or​ false? A. The statement is true. The row reduction algorithm is only useful when it is used to find the solution of a linear system. 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 false. It is possible to create a linear system such that the row reduction algorithm does not apply to the corresponding augmented matrix. D. The statement is true. Every matrix with at least two columns can be interpreted as the augmented matrix of a linear system.

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.

d. If y is in a subspace​ W, then the orthogonal projection of y onto W is y itself. Choose the correct answer below. A. The statement is false because if y is in​ W, then projWy is orthogonal to y​, and is in W⊥. B. The statement is true because for an orthogonal basis of​ W, B=u1,...,up​, y and projWy can be written as linear combinations of vectors in B with equal weights. C. The statement is true because if y is in​ W, then projWy=−y​, which is in the same spanning set as y. D. The statement is false because if y is in​ W, then projWy=0​, so this statement is false unless y=0.

B. The statement is true because for an orthogonal basis of​ W, B=u1,...,up​, y and projWy can be written as linear combinations of vectors in B with equal weights.

c. If AP=​PD, with D​ diagonal, then the nonzero columns of P must be eigenvectors of A. A. The statement is false. AP=PD cannot imply that A is​ diagonalizable, so the columns of P may not be eigenvectors of A. B. The statement is true. Let v be a nonzero column in P and let λ be the corresponding diagonal element in D. Then AP=PD implies that Av=λv​, which means that v is an eigenvector of A. C. The statement is false. If P has a zero​ column, then it is not linearly independent and so A is not diagonalizable. D. The statement is true. AP=PD implies that the columns of the product PD are eigenvalues that correspond to the eigenvectors of A.

B. The statement is true. Let v be a nonzero column in P and let λ be the corresponding diagonal element in D. Then AP=PD implies that Av=λv​, which means that v is an eigenvector of A.

a. If z is orthogonal to u1 and u2 and if W=Span u1,u2​, then z must be in W⊥. Choose the correct answer below. A. The statement is false​ because, since z is orthogonal to u1 and u2​, it exists in Span u1,u2. Since W=Span u1,u2​, z is in W and cannot be in W⊥. B. The statement is true​ because, since z is orthogonal to u1 and u2​, it is orthogonal to every vector in Span u1,u2​, a set that spans W. C. The statement is true because W⊥ is the set of all vectors orthogonal to u1 and u2​, so by​ definition, z is in W⊥. D. The statement is false because if z is orthogonal to u1 and u2​, it only follows that z orthogonal to Span u1 and Span u2. This is not enough information to conclude that z is in W⊥.

B. The statement is true​ because, since z is orthogonal to u1 and u2​, it is orthogonal to every vector in Span u1,u2​, a set that spans W.

e. If B is an echelon form of a matrix​ A, then the pivot columns of B form a basis for Col A. A. This statement is false. The pivot columns of B form a basis for Col A only when B is in reduced row echelon form. B. This statement is false. The columns of an echelon form of a matrix are often not in the column space of the original matrix. C. This statement is true. This is the definition of a column space. D. This statement is false. The pivot columns of B form a basis for Nul A.

B. This statement is false. The columns of an echelon form of a matrix are often not in the column space of the original matrix.

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. True. If the columns of A span ℝm​, then the equation Ax=b has a solution for each b in ℝm. C. False. If the columns of A span ℝm​, then the equation Ax=b is inconsistent for each b in ℝm. D. False. Since the columns of A span ℝm​, the matrix A has a pivot position in exactly m−1 rows.

B. True. If the columns of A span ℝm​, then the equation Ax=b has a solution for each b in ℝm.

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. False. Vector z cannot be in ​Span{x​, y​} because x and y are linearly independent. B. 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​}. 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​}.

B. 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​}.

c. Is the statement​ "Two equivalent linear systems can have different solution​ sets" true or​ false? Explain. A. ​True, because equivalent linear systems are systems with the same number of​ variables, which means that they can have different solution sets. B. ​False, because two systems are called equivalent if they have the same solution set. C. ​False, because two systems are called equivalent only if they both have no solution. D. ​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.

B. ​False, because two systems are called equivalent if they have the same solution set.

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 if Ax=b has at least one solution for each b in ℝn​, then the matrix is not invertible. 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. ​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. D. ​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.

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. An example of a linear combination of vectors v1 and v2 is the vector 12v1. Choose the correct answer below. A. False. Neither of the weights of v1 and v2 can be 0. B. ​True, as 12v1=12v1+0v2. C. ​False, as 12v1 is a multiple of v1 but is independent of v2. D. ​True, as 12v1=14v1+14v2.

B. ​True, as 12v1=12v1+0v2.

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 elementary row operations are always applied to an augmented matrix after the solution has been found. B. ​True, because the elementary row operations replace a system with an equivalent system. C. ​False, because the elementary row operations make a system inconsistent. D. ​False, because the elementary row operations augment the number of rows and columns of a matrix.

B. ​True, because the elementary row operations replace a system with an equivalent system.

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. ​True, by definition of invertible.

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. ​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. 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. ​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.

B. ​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.

d. If the equation Ax=0 has a nontrivial​ solution, then A has fewer than n pivot positions. A. ​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. B. ​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. C. ​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. D. ​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. ​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. If A is an invertible n×n ​matrix, then the equation Ax=b is consistent for each b in ℝn. A. ​True; since A is​ invertible, A−1b=x for all x in ℝn. Multiply both sides by A and the result is Ax=b. B. ​True; since A is​ invertible, A−1b exists for all b in ℝn. Define x=A−1b. Then Ax=b. C. ​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. D. ​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 exists for all b in ℝn. Define x=A−1b. Then Ax=b.

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. ​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; 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. D. ​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.

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

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 the columns are linearly​ independent, then Ax=0 has the trivial solution. 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 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.

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. True. The pivot positions in the augmented matrix Ab always occur in the columns that represent A. B. 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. 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. False. The augmented matrix Ab cannot have a pivot position in every row because it has more columns than rows.

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.

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. True. If A=2304​, then det A=8. C. False. Reduction to an echelon form may also include scaling a row by a nonzero​ constant, which can change the value of the determinant. D. False. The determinant is the product of the number of pivots and ​(−​1)r.

C. False. Reduction to an echelon form may also include scaling a row by a nonzero​ constant, which can change the value of the determinant.

d. The equation x=p+tv describes a line through v parallel to p. A. 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. B. 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. 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. D. 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.

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. False. The equation Ax=b is referred to as a matrix equation because A is a matrix. D. True. The equation Ax=b is referred to as a vector equation because it consists of scalars multiplied by vectors.

C. False. The equation Ax=b is referred to as a matrix equation because A is a matrix.

b. For any scalar​ c, cv=cv. Choose the correct answer below. A. The given statement is true​ because, for v in ℝn​, cv12+cv22+••• +cvn2=cv21+v21+••• +v2n for any value of c. B. The given statement is true because of the Pythagorean Theorem. C. The given statement is false. Since length is always​ positive, the value of cv will always be positive. By the same​ logic, when c is​ negative, the value of cv is negative. Your answer is correct. D. The given statement is false. Since there is a square root involved in the formula for​ length, the value of cv will always be lesser in magnitude than the value of cv.

C. The given statement is false. Since length is always​ positive, the value of cv will always be positive. By the same​ logic, when c is​ negative, the value of cv is negative.

​(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 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. 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 false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. 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.

C. 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.

d. If y is in a subspace​ W, then the orthogonal projection of y onto W is y itself. Choose the correct answer below. A. The statement is false because if y is in​ W, then projWy=0​, so this statement is false unless y=0. B. The statement is true because if y is in​ W, then projWy=−y​, which is in the same spanning set as y. C. The statement is true because for an orthogonal basis of​ W, B=u1,...,up​, y and projWy can be written as linear combinations of vectors in B with equal weights. D. The statement is false because if y is in​ W, then projWy is orthogonal to y​, and is in W⊥.

C. The statement is true because for an orthogonal basis of​ W, B=u1,...,up​, y and projWy can be written as linear combinations of vectors in B with equal weights.

b. For each y and each subspace​ W, the vector y−projWy is orthogonal to W. Choose the correct answer below. A. The statement is false because y can be written uniquely in the form y=projWy+z where z is in W and projWy is in W⊥ and it follows that z=y−projWy. B. The statement is true because y and projWy are both orthogonal to W.​ Thus, a linear combination of them must also be orthogonal to W. C. The statement is true because y can be written uniquely in the form y=projWy+z where projWy is in W and z is in W⊥ and it follows that z=y−projWy. D. The statement is false because y−projWy is in W and so cannot be orthogonal to W.

C. The statement is true because y can be written uniquely in the form y=projWy+z where projWy is in W and z is in W⊥ and it follows that z=y−projWy

a. If z is orthogonal to u1 and u2 and if W=Span u1,u2​, then z must be in W⊥. Choose the correct answer below. A. The statement is false​ because, since z is orthogonal to u1 and u2​, it exists in Span u1,u2. Since W=Span u1,u2​, z is in W and cannot be in W⊥. B. The statement is true because W⊥ is the set of all vectors orthogonal to u1 and u2​, so by​ definition, z is in W⊥. C. The statement is true​ because, since z is orthogonal to u1 and u2​, it is orthogonal to every vector in Span u1,u2​, a set that spans W. D. The statement is false because if z is orthogonal to u1 and u2​, it only follows that z orthogonal to Span u1 and Span u2. This is not enough information to conclude that z is in W⊥.

C. The statement is true​ because, since z is orthogonal to u1 and u2​, it is orthogonal to every vector in Span u1,u2​, a set that spans W.

a. A subset H of ℝn is a subspace if the zero vector is in H. A. This statement is true. This is the definition of a subspace. B. This statement is false. For each u and v in​ H, the product uv must also be in H. C. This statement is false. For each u and v in H and each scalar​ c, the sum u+v and the vector cu must also be in H. D. This statement is false. The subset H is a subspace if the zero vector is not in H.

C. This statement is false. For each u and v in H and each scalar​ c, the sum u+v and the vector cu must also be in H.

b. Given vectors v1​, ​..., vp in ℝn​, the set of all linear combinations of these vectors is a subspace of ℝn. A. This statement is false. This set does not contain the zero vector. B. This statement is false. This set is a subspace of ℝp. C. This statement is true. This set satisfies all properties of a subspace. D. This statement is false. This set is a subspace of ℝn+p.

C. This statement is true. This set satisfies all properties of a subspace.

c. If the columns of A are linearly​ dependent, then det A=0. A. False. If det A=​0, then A is invertible. B. False. The columns of I are linearly​ dependent, yet det I =1. C. True. If the columns of A are linearly​ dependent, then A is not invertible. D. True. If the columns of A are linearly​ dependent, then one of the columns is equal to another.

C. True. If the columns of A are linearly​ dependent, then A is not invertible.

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. False. If the equation Ax=b has infinitely many​ solutions, then the vector b cannot be a linear combination of the columns of A. B. True. The equation Ax=b is unrelated to whether the vector b is a linear combination of the columns of a matrix A. C. True. The equation Ax=b has the same solution set as the equation x1a1+x2a2+•••+xnan=b. D. 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+•••+xnan=b.

d. The first entry in the product Ax is a sum of products. Choose the correct answer below. A. True. The first entry in Ax is the sum of the products of corresponding entries in x and the first column of A. B. False. The first entry in Ax is the product of x1 and the column a1. C. 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. D. False. The first entry in Ax is the sum of the corresponding entries in x and the first entry in each column of A.

C. 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

Is the statement​ "Two matrices are row equivalent if they have the same number of​ rows" true or​ false? Explain. A. ​False, because if two matrices are row equivalent it means that they have the same number of row solutions. 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. D. ​True, because two matrices that are row equivalent have the same number of​ solutions, which means that they have the same number of rows.

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.

a. If the equation Ax=0 has only the trivial​ solution, then A is row equivalent to the n×n identity matrix. A. ​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. ​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. 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.​ 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. ​det(A+​B)=det A+det B A. False. ​det(A+​B)=​(det ​A)(det B) B. True. If A=2010 and B=3050​, then ​det(A+​B)=0 and det A+det B=0. C. True. Determinants are linear transformations. D. False. If A=1001 and B=−100−1​, then ​det(A+​B)=0 and det A+det B=2.

D. False. If A=1001 and B=−100−1​, then ​det(A+​B)=0 and det A+det B=2.

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. 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. C. False. The homogeneous equation Ax=0 never has the trivial solution. D. False. The homogeneous equation Ax=0 always has the trivial solution.

D. False. The homogeneous equation Ax=0 always has the trivial solution.

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. True. The equation Ax=b is always consistent and there always exists a vector p that is a solution. B. 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. C. 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. D. 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.

D. 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.

e. The set Span ​{u​,v​} is always visualized as a plane through the origin. Choose the correct answer below. A. True. The set Span ​{u​,v​} is always visualized as a line in ℝ3 that contains u​, v​, and 0. B. False. Although the set Span ​{u​,v​} is always visualized as a​ plane, it is not always through the origin. C. True. The set Span ​{u​,v​} is always visualized as a plane in ℝ3 that contains u​, v​, and 0. D. False. This statement is often​ true, but Span ​{u​,v​} is not a plane when v is a multiple of u or when u is the zero vector.

D. False. This statement is often​ true, but Span ​{u​,v​} is not a plane when v is a multiple of u or when u is the zero vector.

Could a set of n vectors in ℝm span all of ℝm when n is less than​ m? Explain. Choose the correct answer below. A. No. Without knowing values of n and​ m, there is no way to determine if n vectors in ℝm will span all of ℝm. B. Yes. Any number of vectors in ℝm will span all of ℝm. C. Yes. A set of n vectors in ℝm can span ℝm 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 ℝm. D. 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).

D. 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).

c. If x is orthogonal to every vector in a subspace​ W, then x is in W⊥. Choose the correct answer below. A. The given statement is false. A vector x is in W⊥ if and only if x is orthogonal to every vector in a set that spans W. B. The given statement is false. If x is orthogonal to every vector in a subspace​ W, then x is in​ W, so x cannot be in W⊥. Your answer is not correct. C. The given statement is true. If x is orthogonal to every vector in a subspace​ W, then x=0. ​Thus, x is in W⊥. D. The given statement is true. If x is orthogonal to every vector in​ W, then x is said to be orthogonal to W. The set of all vectors x that are orthogonal to W is denoted W⊥. This is the correct answer.

D. The given statement is true. If x is orthogonal to every vector in​ W, then x is said to be orthogonal to W. The set of all vectors x that are orthogonal to W is denoted W⊥.

a. u•v−v•u=0 Choose the correct answer below. A. The given statement is false. When u and v are​ orthogonal, u•v=​1, so in that​ case, u•v−v•u≠0. B. The given statement is false. When u and v are​ orthogonal, u•v=​0, so in that​ case, u•v−v•u≠0. C. The given statement is true. Since the inner product is​ commutative, u•v=1−v•u. Subtracting v•u from each side of this equation gives u•v−v•u=0. D. The given statement is true. Since the inner product is​ commutative, u•v=v•u. Subtracting v•u from each side of this equation gives u•v−v•u=0.

D. The given statement is true. Since the inner product is​ commutative, u•v=v•u. Subtracting v•u from each side of this equation gives u•v−v•u=0.

​(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 false. The indicated row corresponds to the equation 5x4=​0, which means the system is consistent. B. 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. C. The statement is true. The indicated row corresponds to the equation 5=0. This equation is a​ contradiction, so the linear system is inconsistent. D. The statement is false. The indicated row corresponds to the equation 5x4=​0, which does not by itself make the system inconsistent.

D. The statement is false. The indicated row corresponds to the equation 5x4=​0, which does not by itself make the system inconsistent.

​(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. A variable that corresponds to a pivot column in the coefficient matrix is called a free​ variable, not a basic variable. B. 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. C. The statement is false. Not every linear system has basic variables. D. The statement is true. It is the definition of a basic variable.

D. The statement is true. It is the definition of a basic variable.

b. If y is a linear combination of nonzero vectors from an orthogonal​ set, then the weights in the linear combination can be computed without row operations on a matrix. A. False. The weights in a linear combination can only be computed without row operations on a matrix if one of the vectors is the zero vector. B. True. For each y in​ W, the weights in the linear combination y=c1u1+•••+cpup can be computed by cj=y•ujy•y​, where j=​1, . . .​ , p. C. False. The weights in any linear combination can only be computed using row operations. D. True. For each y in​ W, the weights in the linear combination y=c1u1+•••+cpup can be computed by cj=y•ujuj•uj​, where j=​1, . . .​ , p.

D. True. For each y in​ W, the weights in the linear combination y=c1u1+•••+cpup can be computed by cj=y•ujuj•uj​, where j=​1, . . .​ , p.

a. A row replacement operation does not affect the determinant of a matrix. A. True. Row operations​ don't change the solutions of the matrix equation Ax=b. 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. False. Changing any of the entries in the matrix changes the determinant. D. 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. 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.

c. If the vectors in an orthogonal set of nonzero vectors are​ normalized, then some of the new vectors may not be orthogonal. A. True. Any time an orthogonal set of vectors is normalized then the new set is not orthogonal. B. False. Normalization changes all nonzero vectors to have unit​ length, but does not change their relative angles.​ Therefore, orthogonal vectors will always remain orthogonal after they are normalized. C. False. The normalization process makes the vectors​ orthonormal, but not necessarily orthogonal. D. True. If the original vectors have different​ lengths, then when they are​ normalized, they will be multiplied by different scalars and their inner product will no longer be 0.

D. True. If the original vectors have different​ lengths, then when they are​ normalized, they will be multiplied by different scalars and their inner product will no longer be 0.

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. No. Since A has 2​ pivots, there are no free variables. With no free​ variables, Ax=0 has only the trivial solution. B. 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. C. 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. D. Yes. Since A has 2​ pivots, there is one free variable. So Ax=0 has a nontrivial solution.

D. Yes. Since A has 2​ pivots, there is one free variable. So Ax=0 has a nontrivial solution.

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. ​False; if A is​ invertible, then ab=cd. C. ​True; A=abcd is invertible if and only if a≠c and b≠d. D. ​False; if ad−bc≠​0, then A is invertible.

D. ​False; if ad−bc≠​0, then A is invertible.

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. ​False, because a consistent system has only one unique solution. C. ​True, because a consistent system is made up of equations for planes in​ three-dimensional space. D. ​True, a consistent system is defined as a system that has at least one solution.

D. ​True, a consistent system is defined as a system that has at least one solution.