Question 1

If  has matrix  with respect to bases  for the domain and  for the codomain, then the matrix of T with respect to the bases  and  is  .​

Accepted Answer

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Question 2

If V is a finite-dimensional vector space, then the matrix A of a linear transformation  is invertible if and only if T is one-to-one.

Accepted Answer

This statement is known as the invertible matrix theorem for linear transformations, which states that a linear transformation (or matrix) is invertible if and only if it is both injective (one-to-one) and surjective (onto). Since a matrix is invertible if and only if its corresponding linear transformation is invertible, the matrix A of a linear transformation T is invertible if and only if T is one-to-one.

Question 3

Suppose V and W are finite dimensional vector spaces, and  and  are linear transformations such that  for every v in V and  for every w in W. If the matrices  ,  represent  ,  respectively (with respect to the same bases for V and W), then  .

Accepted Answer

This is a statement of the Rank-Nullity theorem, which states that for any linear transformation f: V → W, the rank of f plus the nullity of f equals the dimension of V. Here, the statement implies that the rank of T plus the nullity of S equals the dimension of V, which is true.

Question 4

If  is a linear transformation, with V a vector space having basis  , and if  for all i, where  is a scalar, then the matrix of T is diagonal, where G is the basis used for both the domain and codomain.​

Accepted Answer

By the given conditions, the linear transformation T maps each basis vector of V to a scalar multiple of itself. This implies that the matrix of T with respect to any basis of V (including the given basis) is diagonal.

Question 5

Let V be a vector space with basis  , and let  be the linear transformation  . Then T is an isomorphism, and the matrix of T with respect to G and the standard basis is the  identity matrix.​

Accepted Answer

By definition, an isomorphism is a bijective linear transformation. Since the basis of the vector space V and the standard basis are related by the identity matrix, and the linear transformation T is bijective, we can conclude that T is an isomorphism.

Question 6

Find the change of basis matrix from G to H.

Accepted Answer

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Question 7

Find the change of basis matrix from G to H.

Accepted Answer

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Question 8

Suppose B is the matrix of  with respect to a basis H, and S is the change of basis matrix from a basis G to H. Find the matrix A of T with respect to G.

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Question 9

Suppose B is the matrix of  with respect to a basis H, and S is the change of basis matrix from a basis G to H. Find the matrix A of T with respect to G.  ​

Accepted Answer

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Question 10

Suppose B is the matrix of  with respect to the basis H. Find the matrix A of T with respect to the basis G.

Accepted Answer

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Question 11

Suppose B is the matrix of  with respect to the basis H. Find the matrix A of T with respect to the basis G.

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Question 12

Determine if A and B are similar matrices.

Accepted Answer

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Question 13

Determine if A and B are similar matrices.
  ,

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Question 14

Determine if A and B are similar matrices.
  ,

Accepted Answer

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Question 15

Determine if A and B are similar matrices.
  ,

Accepted Answer

The answer of Determine if A and B are similar...

Question 16

If A and B are similar matrices and B and C are similar matrices, then A and C are similar matrices.

Accepted Answer

This is a property of similarity. If two matrices are similar to a third matrix, they are also similar to each other. Therefore, if A is similar to B and B is similar to C, then A and C must also be similar to each other.

Question 17

Every change of basis matrix is invertible.

Accepted Answer

Every change of basis matrix is invertible because it is derived from a basis that is linearly independent and spans the vector space, thus ensuring that the columns of the matrix are also linearly independent and span the vector space, making it invertible.

Question 18

If S is a change of basis matrix from basis G to basis H, then  is a change of basis matrix from H to

Accepted Answer

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Question 19

If A is an  matrix, then  is similar to an  diagonal matrix.

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Question 20

If A, B, C, and D are  matrices such that A is similar to B and C is similar to D, then AC is similar to BD.

Accepted Answer

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Quiz 9: Linear Transformations

If has matrix with respect to bases for the domain and for the codomain, then the matrix of T with respect to the bases and is .​

If V is a finite-dimensional vector space, then the matrix A of a linear transformation is invertible if and only if T is one-to-one.

Suppose V and W are finite dimensional vector spaces, and and are linear transformations such that for every v in V and for every w in W. If the matrices , represent , respectively (with respect to the same bases for V and W), then .

If is a linear transformation, with V a vector space having basis , and if for all i, where is a scalar, then the matrix of T is diagonal, where G is the basis used for both the domain and codomain.​

Let V be a vector space with basis , and let be the linear transformation . Then T is an isomorphism, and the matrix of T with respect to G and the standard basis is the identity matrix.​

Find the change of basis matrix from G to H.

Find the change of basis matrix from G to H.

Suppose B is the matrix of with respect to a basis H, and S is the change of basis matrix from a basis G to H. Find the matrix A of T with respect to G.

Suppose B is the matrix of with respect to a basis H, and S is the change of basis matrix from a basis G to H. Find the matrix A of T with respect to G. ​

Suppose B is the matrix of with respect to the basis H. Find the matrix A of T with respect to the basis G.

Suppose B is the matrix of with respect to the basis H. Find the matrix A of T with respect to the basis G.

Determine if A and B are similar matrices.

Determine if A and B are similar matrices. ,

Determine if A and B are similar matrices. ,

Determine if A and B are similar matrices. ,

If A and B are similar matrices and B and C are similar matrices, then A and C are similar matrices.

Every change of basis matrix is invertible.

If S is a change of basis matrix from basis G to basis H, then is a change of basis matrix from H to

If A is an matrix, then is similar to an diagonal matrix.

If A, B, C, and D are matrices such that A is similar to B and C is similar to D, then AC is similar to BD.

If has matrix with respect to bases for the domain and for the codomain, then the matrix of T with respect to the bases and is .

If is a linear transformation, with V a vector space having basis , and if for all i, where is a scalar, then the matrix of T is diagonal, where G is the basis used for both the domain and codomain.

Let V be a vector space with basis , and let be the linear transformation . Then T is an isomorphism, and the matrix of T with respect to G and the standard basis is the identity matrix.

Suppose B is the matrix of with respect to a basis H, and S is the change of basis matrix from a basis G to H. Find the matrix A of T with respect to G.