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\begin{document}
\title{314, S'09-QUIZ 4}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle

\section{$<12pts >$}

Let ${\bf B}$ and ${\bf C}$ be fixed $n\times n$ matrices. Determine whether the
following are linear operators on ${\bf R}^{n\times n}$.
\begin{enumerate}
\item (4 pts) $L(A) = C^2 A + A B$. 
\item (4 pts) $L(A) = C^2 A + B$. 
\item (4 pts) $L(A) = C^2 A B$. \\[.1in]
\end{enumerate}

\noindent {\bf Solution}\\

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\newpage

\section{$< 16 pts >$}
Given the vectors ${\bf u}_1,{\bf u}_2,{\bf u}_3 \in {\bf R}^3$
\[ 
{\bf u}_1 =
\left[
\begin{array}{r} 1 \\ -1 \\ 1 \end{array}
\right] \ ,\ 
{\bf u}_2 =
\left[
\begin{array}{r} -1 \\  2 \\ 0 \end{array}
\right] \ ,\ 
{\bf u}_3 =
\left[
\begin{array}{r} -2 \\  0 \\ 1 \end{array}
\right] 
\]
and the vectors ${\bf v}_1,{\bf v}_2 \in {\bf R}^2$,
\[
{\bf v}_1 =
\left[
\begin{array}{r} 1 \\ -1 \end{array}
\right] \ ,\
{\bf v}_2 =
\left[
\begin{array}{r} 1 \\  1 \end{array}
\right] 
\]
and let ${\cal L}$ be the linear transformation from ${\bf R}^3$ to ${\bf R}^2$
given by
\[ (y_1, y_2)^T = {\cal L}(x_1, x_2, x_3) = (x_2-x_1, x_3-x_1)^T \ .\]
\begin{enumerate}
\item (5pts) Find a matrix ${\bf A}$ such that ${\bf y} = {\cal L}({\bf x}) 
= {\bf A} {\bf x}$
for all inputs ${\bf x}\in {\bf R}^3$ written in the standard basis 
$E_3 = ({\bf e}_1,{\bf e}_2,{\bf e}_3)$ and giving output
${\bf y} \in {\bf R}^2$ in terms of its components in the standard basis 
$E_2 = ({\bf e}_1,{\bf e}_2)$. 
\item (8pts) Find a matrix ${\bf B}$ such that ${\cal L}({\bf x}) = {\bf B} {\bf x} 
={\bf y}$
for all input vectors ${\bf x}\in {\bf R}^3$ written in the basis 
$U = ({\bf u}_1,{\bf u}_2,{\bf u}_3)$ and giving output 
${\bf y} \in {\bf R}^2$ in terms of its components in the basis
$V = ({\bf v}_1,{\bf v}_2)$.  
\item (3pts) Are the matrices ${\bf A}$ and ${\bf B}$ related? How? \\[.1in]
\end{enumerate}
{\bf Solution}\\

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\newpage
\section{$< 12pts >$}
\begin{enumerate}
\item (2pts) Show that the following transformation from $P^2$ to $P^3$ is linear: 
 ${\cal L}(p(x)) = p(1) + \int_0^x p(x) dx \ $.  \\[.1in]
\item (10pts) Write the above linear transformation in terms of a $2\times 3$
matrix if we use the ordered basis $U = \left\{ 1-x, 1+x\right\}$ for $P^2$ and
the standard ordered basis $E = \left\{1, x, x^2\right\}$ for $P^3$.
That is, write ${\cal L}({\bf p}) = {\bf q}$ as ${\bf A}{\bf p}_U = {\bf q}_E$ 
where ${\bf p}(x) = a_1 (1-x) + a_2 (1+x) \Rightarrow
{\bf p}_U = (a_1, a_2)^T$; ${\bf q}(x)= b_1 + b_2 x + b_3 x^2  \Rightarrow
{\bf q}_E = (b1, b_2, b_3)^T$
and
\[ A
\left[
\begin{array}{r} a_1 \\  a_2 \end{array}
\right] =
\left[
\begin{array}{r} b_1 \\  b_2 \\ b_3 \end{array}
\right] \ . 
\]
%\begin{enumerate}
%\item ${\cal L}(p(x)) = p'(x) + \int_0^x p^2(x) dx \ $.
%\item ${\cal L}(p(x)) = log(p^2(x)) + \int_0^x p(x) dx \ $.
%\item ${\cal L}(p(x)) = 1 + \int_0^x p(x) dx \ $.
%\item ${\cal L}(p(x)) = p(1) + \int_0^x p(x) dx \ $.  \\[.1in]
\end{enumerate}
\noindent {\bf Solution}\\
\end{document}
