Homework 6

1. Suppose $A:V\to V$ is a linear transformation of a vector space $V$ over a field $\F$. For a given number $\lambda\in \F$ we consider the set

$L=\bigl\{x\in V \mid \text{$x$ is an eigenvector with eigenvalue $\lambda$}\bigr\}\cup\{0\}$

Write a careful proof of the fact that $L$ is a linear subspace of $V$.

2. Suppose $A:V\to V$ is a linear transformation that satisfies $A^4=O$, and let $\lambda\in\F$ be an eigenvalue of $A$. Show that $\lambda=0$.

In the following problems assume $\F=\R$

3. (a) Compute the eigenvalues of

$A = \mat 0 & 3 & 12 & -4\\ 0 & 0 & -372 & -24\\ 0 & 0 & 0 & -4\\ 0 & 0 & 0 & 0 \rix$

(b) For each eigenvalue, compute all eigenvectors

(c) Does $\R^4$ have a basis consisting of eigenvectors of $A$?

4. (a) Compute the eigenvalues of

$B=\mat 7 & 1 & 24 \\ 0 & 1 & 0 \\ -2& 0 & -7 \rix$

(b) For each eigenvalue, compute all eigenvectors

(c) Does $\R^3$ have a basis consisting of eigenvectors of $B$?