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

\vspace{2 cm}

\centerline{\large{$\mathcal{EXAM}$ 2}}

\bigskip

\centerline{\large{\it November $18^{\mbox{th}}$, 2015}}

\bigskip\bigskip\bigskip

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

\bigskip\bigskip

\noindent\textbf{Directions:}

\bigskip

a. You may NOT use your book or your notes.

b. Please ask for extra scrap paper if needed.

c. Show \underline{all} work. Unless otherwise noted, a solution without work
is worth nothing.

d. The total possible points are 105, but your score will be counted out of 100.

e. Good Luck!

\vspace{.8 in}

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

\bigskip

\begin{center}%
\begin{tabular}
[c]{ll}%
1. & \underline{\hspace{1in}}\\
& \\
2. & \underline{\hspace{1in}}\\
& \\
3. & \underline{\hspace{1in}}\\
& \\
4. & \underline{\hspace{1in}}\\
& \\
5. & \underline{\hspace{1in}}\\
& \\
Total & \underline{\hspace{1in}/100}%
\end{tabular}

\end{center}

\newpage\noindent\textbf{1. (20pts) } Suppose $A$ is a $10\times10$ matrix
with entries in $\mathbb{C}$ and the rank of $A$ is $7.$ For the following,
determine whether the statement is necessarily true, necessarily false, or
there is not enough information to determine if it is true or false. Give
short justifications.

\bigskip

\noindent\textbf{a.} $\det A=0\vspace{1.5in}$

\noindent\textbf{b.} The reduced row echelon form of $A$ contains at least one
row of zeroes.

$\vspace{1.5in}$



\noindent\textbf{c.} There exists an invertible matrix $Q$ such that
$Q^{-1}AQ$ is diagonal.

$\vspace{1.5in}$

\noindent\textbf{d.}  There exist invertible matrices $Q$ and $P$ such that
$QAP$ is diagonal.

$\vspace{1.5in}$

\noindent\textbf{e.}  $A$ is a product of elementary matrices. 

\bigskip\newpage

\noindent\textbf{2. (20pts) }Consider the matrix $A=\left(
\begin{array}
[c]{ccccc}%
1 & -3 & -7 & 5 & -4\\
2 & 2 & 2 & 6 & -8\\
-1 & 2 & 5 & -2 & -1\\
1 & 0 & -1 & 1 & 1
\end{array}
\right)  .\ $Its reduced row echelon form is $\left(
\begin{array}
[c]{ccccc}%
1 & 0 & -1 & 0 & 3\\
0 & 1 & 2 & 0 & -1\\
0 & 0 & 0 & 1 & -2\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)  $. Answer the following questions about $A$:

\bigskip

\noindent\textbf{a.} What is the rank of $A$?$\vspace{1.5in}$

\noindent\textbf{b.} Give a basis for the column space of $A$.

$\vspace{1.5in}$

\noindent\textbf{c.} Give a basis for the nullspace of $A$.

$\vspace{1.5in}$

\noindent\textbf{d.} Suppose for some vector $b\in\mathbb{R}^{4},$ $x=\left(
\begin{array}
[c]{c}%
1\\
2\\
3\\
4\\
5
\end{array}
\right)  $ is a solution to $Ax=b.$ Find all solutions of $Ax=b$ in
$\mathbb{R}^{5}.$



\bigskip\newpage

\noindent\textbf{3. (20pts)} 

\bigskip

\noindent\textbf{a. (10pts)} Prove that an upper triangular $n\times n$ matrix
is invertible if and only if all its diagonal entries are nonzero.

\vspace{3in}

\noindent\textbf{b. (10pts)} Suppose a matrix $A$ is diagonalizable, so there
is an invertible matrix $Q$ such that $Q^{-1}AQ$ is diagonal. Show that for
each standard basis element $e_{i},$ the vector $Qe_{i}$ is an eigenvector for
$A.$ (Hint: each $e_{i}$ is an eigenvector for the diagonal matrix.) 



\bigskip

\bigskip\newpage

\noindent\textbf{4. (25pts) }Let $T$ be an invertible linear operator on a
finite-dimensional vector space $V.$ 

\noindent\textbf{a. (15pts)} Prove that the scalar $\lambda$ is an eigenvalue
of $T$ if and only if $\lambda^{-1}$ is an eigenvalue of $T^{-1}.$ Hint:
Notice that $T-\lambda I_{V}=-\lambda T\left(  T^{-1}-\lambda^{-1}%
I_{V}\right)  .$\vspace{3.5in}

\noindent\textbf{b. (10pts)} Prove that the eigenspace of $T$ corresponding to
$\lambda$ is the same as the eigenspace of $T^{-1}$ corresponding to
$\lambda^{-1}$ (Note:\ you may use the result of part a even if you cannot
prove it.)

\bigskip

\bigskip\newpage

\noindent\textbf{5. (20pts)} Recall that an real inner product space is a
vector space $V$ over $\mathbb{R}$ together with a product $\left\langle
\cdot,\cdot\right\rangle $ such that for all $x,y,z\in V$ and $c\in\mathbb{R}$,

\begin{itemize}
\item $\left\langle x+y,z\right\rangle =\left\langle x,z\right\rangle
+\left\langle y,z\right\rangle ,$

\item $\left\langle cx,z\right\rangle =c\left\langle x,z\right\rangle ,$

\item $\left\langle x,z\right\rangle =\left\langle z,x\right\rangle ,$

\item $\left\langle x,x\right\rangle >0$ if $x\neq\vec{0}.$
\end{itemize}

Consider the vector space $\mathbb{R}^{2\times2}$ of two-by-two matrices with
real entries and define a product $\left\langle A,B\right\rangle =tr\left(
B^{T}A\right)  ,$ where $tr$ denotes the trace (recall that $tr\left(
\begin{array}
[c]{cc}%
a & b\\
c & d
\end{array}
\right)  =a+d$). Show that this is an inner product space. You may use the
following facts without proof:

\begin{enumerate}
\item $\mathbb{R}^{2\times2}$ is a vector space over $\mathbb{R}$.

\item The trace is linear: $tr\left(  cA+B\right)  =c~tr\left(  A\right)
+tr\left(  B\right)  $ for all $c\in\mathbb{R}$ and $A,B\in\mathbb{R}%
^{2\times2}.$

\item The trace is invariant under transpose: $tr\left(  A^{T}\right)
=tr\left(  A\right)  $ for all $A\in\mathbb{R}^{2\times2}.$

\item $\left(  AB\right)  ^{T}=B^{T}A^{T}$ for all $A,B\in\mathbb{R}%
^{2\times2}.$
\end{enumerate}


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