Linear Algebra

A homogeneous system of equations can be inconsistent. Choose the correct answer below. 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 ℝ^m. 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 homogeneous equation cannot be written in the form Ax=0​, where A is an m×n matrix and 0 is the zero vector in ℝ^m. Such a system Ax=0 does not have the solution x=0. ​Thus, a homogeneous system of equations cannot be inconsistent. C. 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 ℝ^m. Such a system Ax=0 always has at least one​ solution, namely x=0. ​Thus, a homogeneous system of equations can be inconsistent. D. 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 ℝ^m. Such a system Ax=0 does not have the solution x=0. ​Thus, a homogeneous system of equations can be inconsistent.

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 ℝ^m. Such a system Ax=0 always has at least one​ solution, namely x=0. ​Thus, a homogeneous system of equations cannot be inconsistent.

The weights c1, ... , cp in a linear combination c1v1+ ... + cpvp cannot all be zero. A. False. Setting all the weights equal to zero results in the vector 0. B. False. Setting all the weights equal to zero does not result in the vector 0. C. True. Setting all the weights equal to zero results in the vector 0. D. True. Setting all the weights equal to zero does not result in the vector 0.

A. False. Setting all the weights equal to zero results in the vector 0.

If a set in ℝ^n is linearly​ dependent, then the set contains more vectors than there are entries in each vector. Choose the correct answer below. A. False. There exists a set in ℝ^n that is linearly dependent and contains n vectors. One example is a set in ℝ^2 consisting of two vectors where one of the vectors is a scalar multiple of the other. B. True. There exists a set in ℝ^n that is linearly dependent and contains more than n vectors. One example is a set in ℝ^2 consisting of three vectors where one of the vectors is a scalar multiple of another. C. True. For a set in ℝ^n to be linearly​ dependent, it must contain more than n vectors. D. False. If a set in ℝ^n is linearly​ dependent, then the set contains more entries in each vector than vectors.

A. False. There exists a set in ℝ^n that is linearly dependent and contains n vectors. One example is a set in ℝ^2 consisting of two vectors where one of the vectors is a scalar multiple of the other.

det A^(−1) = ​(−​1) det A A. False. det A ^(−1) = ​(det ​A)^(−1) B. True. det A ^(−1) is the opposite of det A. C. True. A^(−1) is formed from an odd number of row or columns switches. D. False. det A^(−1) = det A

A. False. det A ^(−1) = ​(det ​A)^(−1)

The homogeneous equation Ax = 0 has a nontrivial solution ____? A. If and only if the equation has at least one free variable. B. If the equation has a unique solution.

A. If and only if the equation has at least one free variable.

When is matrix multiplication defined? A. In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix. B. In order for matrix multiplication to be defined, the number of rows in the first matrix must be equal to the number of columns in the second matrix.

A. In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

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. The solution set of Ax=b is always represented as a plane through the origin. C. Yes. Since the solution set of Ax=0 contains the​ origin, the solution set of Ax=b must contain 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.

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.

​T: ℝ^3→ℝ^2​, T(e1​) =​ (1​,4​), T(e2​) =​ (2​,−10​), and T(e3​) =​ (−3​,6​), where e1​, e2​, e3 are the columns of the 3×3 identity matrix. Is the linear transformation​ one-to-one? A. T is not​ one-to-one because the standard matrix A has a free variable. B. T is​ one-to-one because the column vectors are not scalar multiples of each other. C. T is​ one-to-one because ​T(x​)=0 has only the trivial solution. D. T is not​ one-to-one because the columns of the standard matrix A are linearly independent.

A. T is not​ one-to-one because the standard matrix A has a free variable.

The echelon form of a matrix is unique. Choose the correct answer below. 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. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations. 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 true. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.

A. The statement is false. The echelon form of a matrix is not​ unique, but the reduced echelon form is unique.

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. 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. B. 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. C. The statement is false. Each different choice of a free variable produces the same solution of the system. D. The statement is false. A general solution is the result of an inconsistent​ system, which has no particular solution.

A. 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.

A^T+B^T=(A+B)^T Choose the correct answer below. A. The statement is true. The transpose property states that (A+B)^T=A^T+B^T. B. The statement is false. The transpose property is inapplicable here. C. The statement is true. The transpose property for matrices is the same as for algebraic exponents of real numbers. D. The statement is false. The transpose property states that (A+B)^T=A^TB^T.

A. The statement is true. The transpose property states that (A+B)^T=A^T+B^T.

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 = A^T + B^T. B. The statement is false. For matrices A and B whose sizes are appropriate to​ sum, (A+B)^T = A^T − B^T. C. The statement is false. For matrices A and B whose sizes are appropriate to​ sum, (A+B)^T = A^T + B^T. D. The statement is true. This is a generalized statement that follows from the theorem (A+B)^T = A^T − B^T.

A. The statement is true. This is a generalized statement that follows from the theorem (A+B)^T = A^T + B^T.

Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1, a2, a3}. A. 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. B. False. An augmented matrix having a solution does not mean b is in Span{a1, a2, a3}. C. False. If b corresponds to the origin then it cannot be in Span{a1, a2, a3}.

A. 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.

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​, 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. B. 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. C. 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. D. 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.

A. 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.

The columns of the standard matrix for a linear transformation from ℝ^n to ℝ^m are the images of the columns of the n×n identity matrix under T. Choose the correct answer below. A. True. The standard matrix is the m×n matrix whose jth column is the vector T(ej)​, where ej is the jth column of the identity matrix in ℝ^n. B. True. The standard matrix is the identity matrix in ℝ^n. C. False. The standard matrix only has the trivial solution. D. False. The standard matrix is the m×n matrix whose jth column is the vector T(ej)​, where ej is the jth column whose entries are all 0.

A. True. The standard matrix is the m×n matrix whose jth column is the vector T(ej)​, where ej is the jth column of the identity matrix in ℝ^n.

Any list of five real numbers is a vector in ℝ^5. A. True. ℝ^5 denotes the collection of all lists of five real numbers. B. False. A list of five real numbers is a vector in ℝ^6. C. False. A list of five real numbers is a vector in ℝ^n. D. False. A list of numbers is not enough to constitute a vector.

A. True. ℝ^5 denotes the collection of all lists of five real numbers.

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

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

A linear transformation​ T: ℝ^n → ℝ^m always maps the origin of ℝ^n to the origin of ℝ^m. A. ​True; for a linear​ transformation, ​T(0​) is equal to 0. B. ​True; for a linear​ transformation, ​T(0​) does not equal 0. C. ​False; for a linear​ transformation, ​T(0​) does not equal 0. D. ​False; for a linear​ transformation, ​T(0​) is equal to 0.

A. ​True; for a linear​ transformation, ​T(0​) is equal to 0.

A mapping​ T: ℝ^n→ℝ^m is​ one-to-one if each vector in ℝ^n maps onto a unique vector in ℝ^m. Choose the correct answer below. A. True. A mapping T is said to be​ one-to-one if each x in ℝ^n has at least one image for b in ℝ^m. B. False. A mapping T is said to be​ one-to-one if each b in ℝ^m is the image of at most one x in ℝ^n. C. False. A mapping T is said to be​ one-to-one if each b in ℝ^m is the image of at least one x in ℝ^n. D. True. A mapping T is said to be​ one-to-one if each b in ℝ^m is the image of exactly one x in ℝ^n.

B. False. A mapping T is said to be​ one-to-one if each b in ℝ^m is the image of at most one x in ℝ^n.

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. 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. 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. D. 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.

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.

If det A is​ zero, then two rows or two columns are the​ same, or a row or a column is zero. A. True. If A = [2 3 2 3] and B = [1 2 0 0]​, then det A = 0 and det B = 0. B. False. If A = [2 6 1 3]​, then det A = 0 and the rows and columns are all distinct and not full of zeros. C. 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. D. 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.

B. False. If A = [2 6 1 3]​, then det A = 0 and the rows and columns are all distinct and not full of zeros.

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. 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. B. 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. C. True. A pivot position in every row of a matrix indicates an inconsistent system of equations because the augmented column will always be zeros. D. False. If a coefficient matrix A has a pivot position in every​ row, then the equation Ax=b may or may not be consistent.

B. 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.

If A is a 4×3 ​matrix, then the transformation x ↦ Ax maps ℝ^3 onto ℝ^4. Choose the correct answer below. A. True. The the columns of A are linearly independent. B. False. The columns of A do not span ℝ^4. C. False. The columns of A are not linearly independent. D. True. The columns of A span ℝ^4.

B. False. The columns of A do not span ℝ^4.

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. 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. 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. 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. 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.

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.

The homogeneous equation Ax = 0 has a trivial solution ____? A. If and only if the equation has at least one free variable. B. If the equation has a unique solution.

B. If the equation has a unique solution.

Let A be a 3×3 matrix with two pivot positions. Does the equation Ax=b have at least one solution for every possible b​? A. 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. B. No. A has one free variable. To have at least one solution for every possible b​, A cannot have any free variable. C. Yes. Since A has 2​ pivots, there are no free variables. So there is at least one solution for every possible b. D. No. A has one free​ variable, so there will be no solution to the system for any possible b.

B. No. A has one free variable. To have at least one solution for every possible b​, A cannot have any free variable.

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? A. Nothing can be determined about the columns of AB since the entries of A are unknown. B. The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal. 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 will be equal only if the first two rows of A are equal.

B. The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.

​(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 false. The associative law of multiplication for matrices states that ​A(BC)=​(AB)C. C. The statement is true. The associative law of multiplication for matrices states that ​(AB)C=​B(AC). D. The statement is false. The associative law of multiplication for matrices states that ​(AB)C=​B(AC).

B. The statement is false. The associative law of multiplication for matrices states that ​A(BC)=​(AB)C.

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 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 false. The pivot positions in a matrix depend on the location of the pivot column. D. The statement is true. Every pivot position is determined by the positions of the leading entries of a matrix in reduced echelon form.

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.

The equation Ax=b is homogeneous if the zero vector is a solution. Choose the correct answer below. A. 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 ℝ^m. ​Thus, the zero vector is never a solution of a homogeneous system. B. 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 ℝ^m. If the zero vector is a​ solution, then b=Ax=A0=0. 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 ℝ^m. 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 ℝ^m. If the zero vector is a​ solution, then b=Ax=A0=0​, which is false.

B. 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 ℝ^m. If the zero vector is a​ solution, then b=Ax=A0=0.

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. 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. B. 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. C. 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. 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.

B. 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.

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. False. A and x can only be written as a linear combination of vectors if and only if in Ax=b​, b is nonzero. B. True. The matrix A is the matrix of coefficients of the system of vectors. C. True. Ax can be written as a linear combination of vectors because any two vectors can be combined by addition. D. False. A and x cannot be written as a linear combination because the matrices do not have the same dimensions.

B. True. The matrix A is the matrix of coefficients of the system of vectors.

Two vectors are linearly dependent if and only if they lie on a line through the origin. Choose the correct answer below. A. True. Linearly dependent vectors must always intersect at the origin. B. 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. C. False. If two vectors are linearly dependent then the graph of one will be​ orthogonal, or​ perpendicular, to the other. D. 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.

B. 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.

A linear transformation preserves the operations of vector addition and scalar multiplication. A. ​False; The linear transformation ​T(cu+dv​) is not the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are not preserved. B. ​True; The linear transformation ​T(cu+dv​) is the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are preserved. C. ​False; The linear transformation ​T(cu+dv​) is the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are not preserved. D. ​True; The linear transformation ​T(cu+dv​) is not the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are preserved.

B. ​True; The linear transformation ​T(cu+dv​) is the same as ​cT(u​)+​dT(v​) in ℝ^m. ​Therefore, vector addition and scalar multiplication are preserved.

Every matrix transformation is a linear transformation. A. ​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. 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; 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. D. ​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.

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.

The vector v results when a vector u−v is added to the vector v. A. True. Adding u−v to v results in v. B. False. Adding u−v to v results in u−2v. C. False. Adding u−v to v results in u. D. False. Adding u−v to v results in 2v.

C. False. Adding u−v to v results in u.

If A is an m×n matrix whose columns do not span ℝ^m​, then the equation Ax=b is consistent for every b in ℝ^m. Choose the correct answer below. A. False. If the columns of A do not span ℝ^m​, Ax=b cannot be consistent. B. True. If the columns of A do not span ℝ^m​, b may or may not span ℝ^m. C. False. If the columns of A do not span ℝ^m​, then A does not have a pivot position in every​ row, and row reducing [A b] could result in a row of the form [0 0 ••• 0 c]​, where c is a nonzero real number. D. True. If Ax=b is​ consistent, then the rows of A must span ℝ^m .

C. False. If the columns of A do not span ℝ^m​, then A does not have a pivot position in every​ row, and row reducing [A b] could result in a row of the form [0 0 ••• 0 c]​, where c is a nonzero real number.

If three row interchanges are made in​ succession, then the new determinant equals the old determinant. A. True. If n row interchanges​ happen, then the determinant must be multiplied by (−1)^(n−1). B. True. Row interchanges do not affect the determinant. C. False. If three row interchanges are made in​ succession, then the new determinant equals the negative of the old determinant. D. False. If three row interchanges are made in​ succession, then the new determinant equals (1/3) the old determinant.

C. False. If three row interchanges are made in​ succession, then the new determinant equals the negative of the old determinant.

The standard matrix of a horizontal shear transformation from ℝ^2 to ℝ^2 has the form [a 0] Choose the correct answer below. [0 d] A. False. The standard matrix has the form [1 0] [0 k]. B. False. The standard matrix has the form [1 0] [0 −1]. C. False. The standard matrix has the form [1 k] [0 1]. D. True.

C. False. The standard matrix has the form [1 k] [0 1].

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​} will not contain the origin. B. True. ​Span{u​,v​} is the set of all scalar multiples of u and all scalar multiples of v. C. False. ​Span{u​,v​} includes linear combinations of both u and v.

C. False. ​Span{u​,v​} includes linear combinations of both u and v.

​T: ℝ^3→ℝ^2​, T(e1​) =​ (1​,4​), T(e2​) =​ (2​,−10​), and T(e3​) =​ (−3​,6​), where e1​, e2​, e3 are the columns of the 3×3 identity matrix. Is the linear transformation​ onto? A. T is not onto because the standard matrix A contains a row of zeros. B. T is not onto because the columns of the standard matrix A span ℝ^2. C. T is onto because the columns of the standard matrix A span ℝ^2. D. T is onto because the standard matrix A does not have a pivot position for every row.

C. T is onto because the columns of the standard matrix A span ℝ^2.

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. A. The statement is true. The definition of AB states that each column of AB is a linear combination of the columns of B using weights from the corresponding column of A. B. The statement is true. The definition of AB states that each column of AB is a linear combination of the columns of A using weights from the corresponding rows of B. C. 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. D. 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 rows of B.

C. 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.

(AB)^T=A^TB^T Choose the correct answer below. A. The statement is true. Matrix multiplication is not commutative so the products must remain in the same order. B. 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=A^TB^T. C. 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=B^TA^T. D. 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=A^TB.

C. 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=B^TA^T.

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 true. Every row and column of AB is the corresponding row and column of A multiplied on the right by B. B. The statement is false. The second row of AB is the second row of A multiplied on the left by B. C. 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. D. 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.

C. 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.

AB+AC=​A(B+​C) Choose the correct answer below. A. The statement is false. The distributive law for matrices states that ​A(B+​C)=BA+CA. B. The statement is false. The distributive law does not apply to matrix multiplication. C. The statement is true. The distributive law for matrices states that ​A(B+​C)=AB+AC. D. The statement is true. The distributive law for matrices is the same as for real numbers.

C. The statement is true. The distributive law for matrices states that ​A(B+​C)=AB+AC.

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 ℝ^2 or ℝ^3​, 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 ℝ^2 or ℝ^3​, 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 ℝ^2 or ℝ^3​, 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 ℝ^2 or ℝ^3​, the effect of adding p to v is to move v in a direction parallel to the plane through p and 0.

C. True. Given v and p in ℝ^2 or ℝ^3​, the effect of adding p to v is to move v in a direction parallel to the line through p and 0.

The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax=b​, if​ A= [a1 a2 a3]. Choose the correct answer below. A. False. The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax=b if and only if x has the same number of rows as A. B. False. If A is an m×n matrix with columns [a1 a2 ••• an]​, then b cannot be found in ℝ^m​, and the system is inconsistent. C. True. If A is an m×n matrix with columns [a1 a2 ••• an]​, and b is a vector in ℝ^m​, the matrix equation Ax=b has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 ••• an b]. D. True. The linear system whose augmented matrix is [a1 a2 a3 b] 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 [a1 a2 ••• an]​, and b is a vector in ℝ^m​, the matrix equation Ax=b has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 ••• an 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. True. The equation Ax=b has a solution set if and only if A has a pivot position in every row. B. False. Ax=b is only consistent if the values of b are nonzero. C. True. The equation Ax=b has a nonempty solution set if and only if b is a linear combination of the columns of A. D. False. b is only included in the set spanned by the columns of A if Ax=b is inconsistent.

C. True. The equation Ax=b has a nonempty solution set if and only if b is a linear combination of the columns of A.

Every linear transformation from ℝ^n to ℝ^m is a matrix transformation. Choose the correct answer below. A. True. Every matrix transformation spans ℝ^n. B. False. Not every image ​T(x​) is of the form Ax. C. True. There exists a unique matrix A such that T(x)=Ax for all x in ℝ^n. D. False. Not every vector x in ℝ^n can be assigned to a vector ​T(x​) in ℝ^m.

C. True. There exists a unique matrix A such that T(x)=Ax for all x in ℝ^n.

If​ T: ℝ^n → ℝ^m is a linear transformation and if c is in ℝ^m​, then a uniqueness question is​ "Is c in the range of​ T?" A. ​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. B. ​True; the question​ "is c in the range of​ T?" is the same as​ "is c the image of a unique x in ℝ^n​?" This is a uniqueness question. 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. ​False; the question​ "is c in the range of​ T?" is the same as​ "is c the image of a unique x in ℝ^n​?" This is an existence question.

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.

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

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

The range of the transformation x ↦ Ax is the set of all linear combinations of the columns of A. A. ​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. B. ​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. C. ​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. 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.

C. ​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.

The determinant of A is the product of the diagonal entries in A. A. False. The determinant is equal to ad−bc. B. True. The determinant 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. C. True. The determinant is the product of the pivots and the pivots can be found on the diagonal. D. False. This is only true if A is triangular.

D. False. This is only true if A is triangular.

How many rows and columns must a matrix A have in order to define a mapping from ℝ^5 into ℝ^9 by the rule ​T(x​)=Ax​? Choose the correct answer below. A. The matrix A must have 9 rows and 9 columns. B. The matrix A must have 5 rows and 5 columns. C. The matrix A must have 5 rows and 9 columns. D. The matrix A must have 9 rows and 5 columns.

D. The matrix A must have 9 rows and 5 columns.

Whenever a system has free​ variables, the solution set contains many solutions. Choose the correct answer below. A. The statement is true. If a solution has at least one free​ variable, there are infinitely many solutions. B. The statement is false. A system with one free variable will have only one solution. C. The statement is true. If there are infinitely many​ solutions, then there must be at least one free variable. 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.

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.

If A and B are 3×3 matrices and B = [b1 b2 b3]​, 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 the resulting matrix AB is the sum of the columns of A using the weights from the corresponding columns of B. B. 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 = A[b1 b2 ... bp] = [Ab1 + Ab2 + ... + Abp]. C. The statement is false. The matrix [Ab1 + Ab2 + Ab3] is the correct size​ matrix, but the plus signs should be minus signs. D. 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 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.

The transpose of a product of matrices equals the product of their transposes in the same order. Choose the correct answer below. A. The statement is true. Matrix multiplication is not commutative so the product of the transposes must be applied in the same order as the initial product. B. The statement is false. The transpose of a product of matrices equals the sum of the transposes of the matrices. C. The statement is true. The product of the transposes can be applied in any order. D. The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order.

D. The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order.

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 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. 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 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. 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. 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.

Every matrix equation Ax=b corresponds to a vector equation with the same solution set. Choose the correct answer below. A. False. The matrix equation Ax=b does not correspond to a vector equation with the same solution set. B. True. The matrix equation Ax=b is simply another notation for the vector equation x1a1 + x2a2 + ••• + xnan = b​, where a1​, ​... , an are the rows of A. C. False. The matrix equation Ax=b only corresponds to an inconsistent system of vector equations. D. True. The matrix equation Ax=b is simply another notation for the vector equation x1a1 + x2a2 + ••• + xnan = b​, where a1​, ​... , an are the columns of A.

D. True. The matrix equation Ax=b is simply another notation for the vector equation x1a1 + x2a2 + ••• + xnan = b​, where a1​, ​... , an are the columns of A.

Let A be a 3×3 matrix with two pivot positions. Does the equation Ax=0 have a nontrivial​ solution? A. 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. 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. 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.

Is the statement​ "A consistent system of linear equations has one or more​ solutions" true or​ false? Explain. A. ​True, because a consistent system is made up of equations for planes in​ three-dimensional space. B. ​False, because a consistent system has infinitely many solutions. 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.

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

A linear map f: ℝ^n→ℝ^m is 1) one-to-one [if and only if] its matrix has rank n (# of pivots = n) and 2) onto [if and only if] its matrix has rank m (# of pivots = m = # of rows).

Definition

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 ℝ^m. Such a system Ax=0 always has at least one solution, namely, x=0 (the zero vector in ℝ^n).

Definition

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. ​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. C. ​False, because if two matrices are row equivalent it means that they have the same number of row solutions. D. ​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.

​B. 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.