Exam 9: Linear Transformations

Determine if A and B are similar matrices. ,

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

Determine whether the function is a linear transformation, where .

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 .​

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

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

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

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

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

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

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

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 .

Determine if A and B are similar matrices.

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

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

R is isomorphic to the subspace S of defined by .

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 .

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