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\begin{center}
{\large Math 136: Homework 2} \\
Due Thursday, April 5
\end{center}

\begin{enumerate}

\item Exercise 5.3: for any angle $\theta$, write 
\[
T_{\theta} = \begin{pmatrix} 
\cos \theta & - \sin \theta \\ 
\sin \theta & \cos \theta
\end{pmatrix}
\]
for the matrix representing rotation of $\mathbf{R}^{2}$
counterclockwise by angle $\theta$.  Fix angles $\alpha$ and $\beta$
and consider the rotation matrices $T_{\alpha}$, $T_{\beta}$, and
$T_{\alpha + \beta}$.

\begin{enumerate}
\item Compute the matrix product $T_{\alpha} T_{\beta}$.
\item Explain, geometrically, why $T_{\alpha} T_{\beta} = T_{\alpha +
\beta}$. 
\item Deduce formulas for $\sin (\alpha + \beta)$ and $\cos (\alpha +
\beta)$.
\end{enumerate}

\item An $n \times n$ matrix $A=(a_{ij})$ is called \emph{upper
triangular} if $a_{ij}=0$ when $i>j$. For example, $\begin{pmatrix}
1 & 3 & 5 \\
0 & 2 & 6 \\
0 & 0 & -9
\end{pmatrix}$ is upper triangular. Prove that the product of two
$n\times n$ upper triangular matrices is upper triangular.

\item Let $V$ denote the set of continuous functions on the interval
$[0,1]$. It is a vector space under the operations of addition of
functions and multiplication by a real number.

\begin{enumerate}
\item Let $S$ be the following subset of $V$:
\[
S = \{f \in V \,\colon\, f(0) = 0 \}.
\]
Does $S$ form a subspace of $V$?  Justify your answer.
\item Let $T$ be the following subset of $V$:
\[
T = \{f \in V \,\colon\, f(0) \neq 0 \}.
\]
Does $T$ form a subspace of $V$?  Justify your answer.
\end{enumerate}

\item Choose the numbers $a$, $b$, $c$, $d$, in the following
augmented matrix so that (a) there is no solution (b) there are
infinitely many solutions to the corresponding system of linear
equations:
\[
\left(
\begin{array}{ccc|c}
 1 & 2 & 3 & a \\
 0 & 4 & 5 & b \\
 0 & 0 & d & c
\end{array}
\right)
\]

\end{enumerate}

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