91Ó°ÊÓ

Prove that if a linear transformation \(T: V \rightarrow W\) is an isomorphism, then there is a linear transformation \(T^{-1}: W \rightarrow V\) satisfying \(\left(T^{-1} \circ T\right)(\mathbf{v})=\mathbf{v}\) for all \(\mathbf{v} \in V\) and \(\left(T \circ T^{-1}\right)(\mathbf{w})=\mathbf{w}\) for all \(\mathbf{w} \in W\)

Prove or give a counterexample: a. If \(B\) is similar to \(A\), then \(B^{\mathrm{T}}\) is similar to \(A^{\mathrm{T}}\). b. If \(B^{2}\) is similar to \(A^{2}\), then \(B\) is similar to \(A\). c. If \(B\) is similar to \(A\) and \(A\) is nonsingular, then \(B\) is nonsingular. d. If \(B\) is similar to \(A\) and \(A\) is symmetric, then \(B\) is symmetric. e. If \(B\) is similar to \(A\), then \(\mathbf{N}(B)=\mathbf{N}(A)\). f. If \(B\) is similar to \(A\), then \(\operatorname{rank}(B)=\operatorname{rank}(A)\).

Show that similarity of matrices is an equivalence relation. That is, verify the following. a. Reflexivity: Any \(n \times n\) matrix \(A\) is similar to itself. b. Symmetry: For any \(n \times n\) matrices \(A\) and \(B\), if \(A\) is similar to \(B\), then \(B\) is similar to A. c. Transitivity: For any \(n \times n\) matrices \(A, B\), and \(C\), if \(A\) is similar to \(B\) and \(B\) is similar to \(C\), then \(A\) is similar to \(C\).