Exam 9: Linear Transformations

Let be a linear transformation satisfying . Find

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

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 the linear transformation is an isomorphism.

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. ​

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

Find the change of basis matrix from G to H.

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

Every change of basis matrix is invertible.

Find for the isomorphism , where

Find the change of basis matrix from G to H.

Determine if A and B are similar matrices. ,

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

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

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.

Let be a linear transformation satisfying .Find .

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

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.​

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

Determine whether the function is a linear transformation, where .