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\begin{center}
{\large Math 136: Homework 5} \\
Due Thursday, May 3
\end{center}

\begin{enumerate}

\item Consider the linear map $T_A: \R^3 \to \R^3$ defined by $\vec{v}
\mapsto A \vec{v}$, where $\displaystyle A = \begin{pmatrix} 1 & -2 &
0 \\ 1 & 4 & 0\\ -2 & -2 & 1 \end{pmatrix}$.  Find the matrix
associated to $T_A$ with respect to the basis
\[
\{\vec{v}_1,\vec{v}_2,\vec{v}_3\} = 
\left\{
\begin{pmatrix}0\\ 0\\ 1 \end{pmatrix},\,
\begin{pmatrix}2\\ -1\\ -2 \end{pmatrix},\,
\begin{pmatrix}-1\\ 1\\ 0 \end{pmatrix}
\right\} .
\]

\item Consider the function 
\[
f(x,y) = 3 x^2 + 2 x y + 3 y^2 = 
\begin{pmatrix} x & y \end{pmatrix}
\begin{pmatrix} 3   & 1 \\ 1 & 3\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} .
\]
Consider the new variables  $(X,Y)$ where
\[
\begin{pmatrix} X \\ Y \end{pmatrix} =
\begin{pmatrix} 
\cos \theta & - \sin \theta \\
\sin\theta & \cos\theta 
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} .
\]
For an appropriate choice of $\theta$, the function $f$ will assume
the form $\displaystyle f(x,y) = a X^2 + b Y^2$.  Find $a$, $b$, and
$\theta$ by diagonalizing the matrix $\displaystyle \begin{pmatrix} 3
& 1 \\ 1 & 3\end{pmatrix}$.

\item Let
$A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 2  & 1\\ 0 & 0 & 3\end{pmatrix}$
and show that $\displaystyle e^A = \begin{pmatrix} e & e^2 - e & e^3 - e^2 \\ 
0 & e^2 & e^3 - e^2 \\ 0 & 0 & e^3 \end{pmatrix}$.

\item Let $I$ denote the $n \times n$ identity matrix, and suppose
that $B$ is an $n \times n$ matrix satisfying $B^{2}=0$ (for example,
$B=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$).  Let $c$ be a
scalar and show that $e^{cI+B} = e^{c} I + e^{c} B$.  (Recall from
class that if $X$ and $Y$ are matrices which commute, then
$e^{X+Y}=e^{X}e^{Y}$.)

\end{enumerate}

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