Page 659 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let A and B be similar matrices. Prove:
9.

(a) and are similar.

(b) If A and B are invertible, then and are similar.

10. (Fredholm Alternative Theorem)

Let be a linear operator on an n-dimensional vector space. Prove that exactly one of the following
statements holds:

(i) The equation has a solution for all vectors b in V.

(ii) Nullity of .

Let be the linear operator defined by
11.

Find the rank and nullity of T.

Prove: If A and B are similar matrices, and if B and C are similar matrices, then A and C are similar matrices.
12.

Let be the linear operator defined by . Find the matrix for L with respect to the standard
13. basis for
.

Let and be bases for a vector space V, and let
14.

be the transition matrix from to B.
(a) Express , , as linear combinations of , , .
(b) Express , , as linear combinations of , , .

Let be a basis for a vector space V, and let be a linear operator such that
15.

Find , where is the basis for V defined by