Exam 9: Linear Transformations

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

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

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

Determine if A and B are similar matrices. ,

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.​

Determine if the linear transformation is an isomorphism.

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

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

Find for the isomorphism , where .

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.​

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

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

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

Find for the isomorphism , where , for .​

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

Determine whether the function is a linear transformation, where .

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

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

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.

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