Chapter 3.4, Problem 71E

Suppose that matrix A is similar to B, with $B=S^{-1}AS$ .
a. Show that if $\stackrel{\to }{x}$ is in ker(B), then $S\stackrel{\to }{x}$ is in ker(A).
b. Show that $\text{nullity}\left(\text{A}\right)=\text{nullity}\left(\text{B}\right)$ . Hint: If ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...},{\stackrel{\to }{v}}_p$ , is a basis of ker(B), then the vectors $S{\stackrel{\to }{v}}_1,S{\stackrel{\to }{v}}_2,\mathrm{...},S{\stackrel{\to }{v}}_p$ in ker(A) are linearly independent. Now reverse the roles of A and B.