% LaTeX file -- M245, Fall 2011, Assignment 1

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 245 \\ Assignment 1\\
Due Friday September 28}
\end{center}
\vskip 2em 

In this assignment, $V$ is always a finite dimensional vector space over a field $\bF$.\\

\begin{Ex}

\item \begin{parts}
\item Find all eigenvalues and eigenspaces for the following matrices ($\bF = \bC$):
\[
\text{(i)}\ \ A = \begin{bmatrix}1&-1\\2&\phantom{-}3\end{bmatrix} \qquad \quad
\text{(ii)}\ \ B = \begin{bmatrix}
\phantom{-}7&-5&-5\\
-5&\phantom{-}2&\phantom{-}0\\
-5&\phantom{-}0&\phantom{-}2
\end{bmatrix}\qquad \quad
\text{(iii)}\ \ C = \begin{bmatrix}-3&3&0\\-5&1&4\\-4&2&2\end{bmatrix}.
\]
\item Which of these matrices can be diagonalized? Which can be diagonalized if $\bF=\bR$?
\end{parts}

\item Suppose that  $V_1,\dots,V_k$ are subspaces such that $V = \sum_{i=1}^k V_i$.
Prove that this is a direct sum if and only if 
\[ \sum_{i=1}^k \dim V_i = \dim V .\]

\item Suppose that $V=V_1 \oplus V_2$. 
\begin{parts}
\item Show that $T \in \L(V)$ can be written as
$T = \begin{bmatrix} T_{11}&T_{12}\\T_{21}&T_{22}\end{bmatrix}$ where \vspace{.3ex}
$T_{ij} \in \L(V_j,V_i)$ for $1 \le i,j \le 2$. 
\hint choose a basis for $V_1$ and $V_2$, combine to get a basis for $V$, 
and decompose the matrix for $T$ in this basis.
\item Suppose that $T = \begin{bmatrix} T_{11}&T_{12}\\0&T_{22}\end{bmatrix}$.
Show that $p_T(x) = p_{T_{11}}(x) p_{T_{22}}(x)$.
\end{parts}


\item Let $A,B \in \L(V)$.
\begin{parts}
\item Show that $AB$ is singular if and only if $BA$ is singular.
\item If $0 \ne \lambda \in \bF$, and $\lambda \not\in \sigma(BA)$, calculate
\[ \big( A (BA-\lambda I)^{-1} B - I \big)(AB-\lambda I) .\]
\item Hence show that $AB$ and $BA$ have the same spectrum.
\end{parts}


\item \textbf{Bonus.}  Let $T \in \L(V)$. Define $L \in \L(\L(V))$ by $L(A) = TA$ for $A \in \L(V)$.
Express $p_L$ in terms of $p_T$.


\end{Ex}
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