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\noindent{\large \textbf{MATH 413/513 \hfill}}

\vspace{2 cm}

\centerline{\large{$\mathcal{FINAL}$ }}

\bigskip

\centerline{\large{\it Due December $17^{\mbox{th}}$, 2015 by 12:30 pm}}

\bigskip\bigskip\bigskip

\noindent\textbf{{\large Your Name:}} \underline{\hspace{5 in}}

\bigskip

\bigskip

\noindent\textbf{{\large I certify that the work on this final is mine and
mine alone (please sign):}}

\underline{\hspace{5in}}\bigskip\bigskip

\vspace{0.8in}

\noindent\textbf{{\large Score:}}

\bigskip

\begin{center}%
\begin{tabular}
[c]{rl}%
1. & \underline{\hspace{1in}}\\
& \\
2a-d. & \underline{\hspace{1in}}\\
& \\
3. & \underline{\hspace{1in}}\\
& \\
2e. (Comprehensive) & \underline{\hspace{1in}}\\
& \\
TCE Extra Credit & \underline{\hspace{0.9in}}3\\
& \\
Total & \underline{\hspace{1in}/100}%
\end{tabular}



\end{center}

\newpage\noindent\textbf{1. (32pts)} Let $V$ and $W$ be finite-dimensional
vector spaces of dimension $n$ and $m$ respectively. Let $T:V\rightarrow W$ be
a linear transformation. Let $\beta=\left\{  v_{1},\ldots,v_{n}\right\}  $ and
$\gamma=\left\{  w_{1},\ldots,w_{m}\right\}  $ be bases for $V$ and $W$ respectively.



\noindent\textbf{a.} Show that if $T$ is one-to-one, then $T\left(
\beta\right)  =\left\{  T\left(  v_{1}\right)  ,\ldots,T\left(  v_{n}\right)
\right\}  $ is linearly independent.

\noindent\textbf{b.} Show that if $T$ is onto, then $T\left(  \beta\right)  $
spans $W.$

\noindent\textbf{c.} Suppose $n\leq m.$ Show that there exists a linear map
$S:W\rightarrow V$ such that $ST=I_{V}$ if and only if $T$ is one-to-one.

\noindent\textbf{d.} If $\left\langle \cdot,\cdot\right\rangle _{V}$ and
$\left\langle \cdot,\cdot\right\rangle _{W}$ are inner products on $V$ and $W$
respectively, show that there exists a linear transformation $T^{\ast
}:W\rightarrow V$ such that
\[
\left\langle T\left(  x\right)  ,y\right\rangle _{W}=\left\langle x,T^{\ast
}\left(  y\right)  \right\rangle _{V}%
\]
for all $x\in V$ and $y\in W.$

\bigskip

\noindent\textbf{2. (32pts)} Let $F$ be a field (you can let it be
$\mathbb{R}$ if this is confusing you). Let $X\in F^{n\times n}$ be a matrix
of unknowns and define the map $T_{v}:F^{n\times n}\rightarrow F^{n}$ by
$T_{v}\left(  X\right)  =Xv$ where $v$ is a fixed (known) vector.

\noindent\textbf{a. }Show that $T_{v}$ is a linear transformation.



\textbf{For parts b, c, and d, let }$n=2$\textbf{ and }$v=\left(
\begin{array}
[c]{c}%
1\\
1
\end{array}
\right)  .$

\noindent\textbf{b.} Find a basis $\beta$ for the vector space of $2\times2$
real-valued matrices (you may use one we have already shown is a basis without
proof) and write $\left[  T_{v}\right]  _{\beta}^{E_{2}}$ where $E_{2}$ is the
standard basis for $\mathbb{R}^{2}.$

\noindent\textbf{c.} Find the nullspace of $T_{v}$.

\noindent\textbf{d.} For which vectors $w\in\mathbb{R}^{2}$ is there a
solution to $T_{v}\left(  X\right)  =w$? What is the dimension of the space of
solutions when there is a solution?

\noindent\textbf{e.} (Comprehensive option only, worth +10pts). If $v=\left(
\begin{array}
[c]{c}%
1\\
\vdots\\
1
\end{array}
\right)  \in\mathbb{R}^{n},$ write $\left[  T_{v}\right]  _{\beta}^{E_{n}}$
for some basis $\beta$ of $\mathbb{R}^{n\times n}.$

\bigskip



\noindent\textbf{3. (40pts)} Consider the space of continuous functions
$\mathcal{F}_{c}\left(  \left[  -\pi,\pi\right]  ,\mathbb{R}\right)  .$ Let
\[
\beta=\left\{  1,\cos x,\sin x,\cos2x,\sin2x\right\}
\]
and let $S=\operatorname{span}\beta$. The vector space $\mathcal{F}_{c}\left(
\left[  -\pi,\pi\right]  ,\mathbb{R}\right)  $ has an inner product
\[
\left\langle f,g\right\rangle =\int_{-\pi}^{\pi}f\left(  x\right)  g\left(
x\right)  dx
\]
and $\beta$ is an orthogonal set with respect to that inner product (you can
check this).

\noindent\textbf{a.} Show that $\dim S=5.$

\noindent\textbf{b.} Show that $\cos^{2}x\in S.$

\noindent\textbf{c.} Show that the derivative $\frac{d^{2}}{dx^{2}}$ is a
linear operator on $S$.

\noindent\textbf{d.} Show that $\frac{d^{2}}{dx^{2}}$ is self-adjoint on $S.$
You may use the orthogonality of $\beta$ (and the integral facts it implies,
such as $\int_{-\pi}^{\pi}\sin x\cos x~dx=0$) without proof.

\noindent\textbf{e.} Write $\left[  \frac{d^{2}}{dx^{2}}\right]  _{\beta}$ and
give the eigenvalues and eigenvectors of $\frac{d^{2}}{dx^{2}}.$

\noindent\textbf{f.} Give the characteristic polynomial of $\frac{d^{2}%
}{dx^{2}}$ considered as a linear operator $S\rightarrow S.$
\end{document}