Deck 9: Linear Transformations

Question 1

Let be a linear transformation satisfying .
Find

Answer 1

Question 2

Let be a linear transformation satisfying .Find .

Answer 2

Question 3

Determine whether the function is a linear transformation, where .

Answer 3

T is a linear transformation.

Question 4

Determine whether the function is a linear transformation, where .

Question 5

Determine whether the function is a linear transformation, where .

Question 6

Determine the kernel and range of the linear transformation given by
.

Question 7

Determine if the linear transformation is one-to-one and/or onto.

Question 8

Let Q be an invertible matrix, and let P be an invertible matrix. Determine whether the function is a linear transformation, where , and if so, determine if T is one-to-one and/or onto.​

Question 9

If S is a nonzero subspace of , determine whether the function is a linear transformation, where , and if so, determine .​

Question 10

If and are linear transformations, then the composition is a linear transformation.

Question 11

Suppose B is an invertible matrix. The function defined by is a one-to-one and onto linear transformation.

Question 12

The function defined by , where and are nonzero scalars, is a linear transformation.

Question 13

Suppose is a linear transformation and is a set of vectors in V. If is a linearly dependent set, then so is .​

Question 14

If is an onto linear transformation, then T is one-to-one.

Question 15

Suppose is a linear transformation that is not one-to-one, and is not the trivial transformation, that is, for some v in V. Then .

Question 16

Suppose S is a proper subspace of . Can S be isomorphic to ?

Question 17

Suppose S is a proper subspace of . Can S be isomorphic to ?

Question 18

Determine if the linear transformation is an isomorphism.

Question 19

Determine if the linear transformation is an isomorphism.

Question 20

Find for the isomorphism , where .

Question 21

Suppose A and B are invertible matrices. Find for the isomorphism , where .

Question 22

Find for the isomorphism , where

Question 23

Suppose are orthogonal matrices, and let be defined by . Verify that is a linear transformation, determine if is an isomorphism, and if so, find .

Question 24

Let be defined by . Verify that is a linear transformation, determine if is an isomorphism, and if so, find .​

Question 25

Find for the isomorphism , where , for .​

Question 26

If and are isomorphisms, then the composition is an isomorphism.

Question 27

If and are isomorphic subspaces of a vector space V then .

Question 28

R is isomorphic to the subspace S of defined by .

Question 29

The subspace S of of all sequences that are eventually zero is isomorphic to .

Question 30

The vector spaces and are isomorphic. (Recall that denotes the vector space of all linear transformations from into

Question 31

Find v given the coordinate vector with respect to the basis G.

Question 32

Find v given the coordinate vector with respect to the basis G.

Question 33

Find the coordinate vector of v with respect to the basis G.

Question 34

Find the coordinate vector of v with respect to the basis G.

Question 35

Suppose A is the matrix of the linear transformation with respect to bases G and Q, respectively. Find for the given .

Question 36

Suppose A is the matrix of the linear transformation with respect to bases G and Q, respectively. Find for the given .

Question 37

Find the matrix A of the linear transformation with respect to bases G and Q, respectively.

Question 38

Find the matrix A of the linear transformation with respect to bases G and Q, respectively.

Question 39

Let . Find the matrix A of the linear transformation with respect to bases G and Q, respectively.
, ​

Question 40

Suppose that has matrix with respect to the basis for the domain and for the codomain. Use the inverse of to find .​

Question 41

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 .​

Question 42

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.

Question 43

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 .

Question 44

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.​

Question 45

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.​

Question 46

Find the change of basis matrix from G to H.

Question 47

Find the change of basis matrix from G to H.

Question 48

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.

Question 49

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. ​

Question 50

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

Question 51

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

Question 52

Determine if A and B are similar matrices.

Question 53

Determine if A and B are similar matrices.
,

Question 54

Determine if A and B are similar matrices.
,

Question 55

Determine if A and B are similar matrices.
,

Question 56

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

Question 57

Every change of basis matrix is invertible.

Question 58

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

Question 59

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

Question 60

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.