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\title{Standard
%TCIMACRO{\TeXButton{LaTeX}{\LaTeX{}} }%
%BeginExpansion
\LaTeX{}
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Article}
\author{A. U. Thor\\The University of Stewart Island}
\maketitle

\begin{abstract}
We study the effects of warm water on the local penguin population. The major
finding is that it is extremely difficult to induce penguins to drink warm
water. The success factor is approximately $-e^{-i\pi}-1$.

\end{abstract}

\begin{center}
{\LARGE LINEAR ALGEBRA PROBLEMS}

{\LARGE UNIVERSITY OF ARIZONA INTEGRATION WORKSHOP}

{\LARGE 2018}
\end{center}

\bigskip

\section{Lecture 1}

\begin{enumerate}
\item Prove that in any vector space $V$, for all $v\in V$, $(-1)v=-v$.

\item Let $F$ be any field. Let $F[x]$, $F[[x]]$ and $F[[x]]_{C}$ be as
defined in the notes. \ Is the set
\[
\{1,x,x^{2},x^{3},\cdots\}
\]
a basis for any of these vector spaces?

\item For $n\in%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$, let
\[
C_{n}=\{f:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\;|\;f(x)\text{ is continuous on }[n,n+1]\text{ and }f(n)=f(n+1)=0\}
\]


\begin{enumerate}
\item Prove that for all $n\in%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$, $C_{n}$ is an $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ vector space.

\item Describe%
\[%
%TCIMACRO{\tprod \limits_{\in\mathcal{\mathbb{Z}}}}%
%BeginExpansion
{\textstyle\prod\limits_{\in\mathcal{\mathbb{Z}}}}
%EndExpansion
C
\]


\item Give an example of something in $%
%TCIMACRO{\tprod \limits_{\in\mathcal{\mathbb{Z}}}}%
%BeginExpansion
{\textstyle\prod\limits_{\in\mathcal{\mathbb{Z}}}}
%EndExpansion
C$ that is not in $%
%TCIMACRO{\tbigoplus \limits_{n\in\mathcal{\mathbb{Z}}}}%
%BeginExpansion
{\textstyle\bigoplus\limits_{n\in\mathcal{\mathbb{Z}}}}
%EndExpansion
C_{n}$.
\end{enumerate}

\item Let $W_{1}$ and $W_{2}$ be subspaces of the finitely generated vector
space $V$ over the field $F$.

\begin{enumerate}
\item Prove that%
\[
\text{Dim}_{F}(W_{1}+W_{2})=\text{Dim}_{F}(W_{1})+\text{Dim}_{F}%
(W_{2})-\text{Dim}_{F}(W_{1}\cap W_{2})
\]


\item Prove that
\[
(W_{1}+W_{2})\cong(W_{1}\oplus W_{2})\text{ \ \ \ }\Leftrightarrow\text{
\ \ \ }W_{1}\cap W_{2}=\{0\}\text{\ }%
\]


\item Is the statement in (b) true or false if we drop the condition that $V$
is finitely generated?
\end{enumerate}

\item Let $V$ and $W$ be vector spaces over $F$ with Dim$_{F}(V)=n$ and
Dim$_{F}(V)=m$. Prove that Hom$_{F}(V,W)$ is a vector space over $F$ of
dimension $nm$. (We haven't seen matrices yet, but that doesn't mean we can't
think of them.)

\item Let $W$ be a subspace of the vector space $V$ over the field $F$. Let
$U$ be a subspaces of the vector space $W$ \ 

\begin{enumerate}
\item Prove that $W/U$ is a subspace of $V/U$.

\item Prove that
\[
\left(  V/U\right)  /W/U)\cong V/W.
\]

\end{enumerate}

\item Let $V$ be a vector space over $F$ with basis $S$. Let%
\[
V^{\ast}=\text{Hom}_{F}(V,F)
\]
For any $s\in S$, let $s^{\ast}:V\rightarrow F$ be given by
\[
s^{\ast}\left(
%TCIMACRO{\tsum \limits_{t\in S}}%
%BeginExpansion
{\textstyle\sum\limits_{t\in S}}
%EndExpansion
a_{t}t\right)  =a_{s}%
\]


\begin{enumerate}
\item Prove that, for all $s\in S$, $s^{\ast}\in$Hom$_{F}(V,F)$.

\item Prove that $S^{\ast}=\{s^{\ast}\in\,$Hom$_{F}(V,F)$\ $|\;s\in S$ $\}$
\ is linearly independent in Hom$_{F}(V,F)$.

\item Show that if $V=F^{n}$ considered as column vectors, then $V^{\ast}$ can
be considered as row vectors in an isomorphic copy of $F^{n}$.
\end{enumerate}

\item Let $F=%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}$ for a prime $p\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$.

\begin{enumerate}
\item How many elements are in the field $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}$?

\item How many elements are in the vector space $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n}$?

\item How many elements are in the vector space Hom$_{%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}}(%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n},%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{m})$?

\item How many elements of Hom$_{%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}}(%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n},%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n})$ have inverses in Hom$_{%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}}(%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n},%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p}^{n})$.
\end{enumerate}

\item Let $V$ and $W$ be vector spaces over $F$. Let $T:V\rightarrow W$ be a
linear transformation. Prove that both the kernel and the image are subspaces
of the respective vector spaces.

\item Let $V$ and $W$ be vector spaces over $F$. Let $T:V\rightarrow W$ be a
linear transformation.

\begin{enumerate}
\item Prove that $T$ is injective if and only if Ker$(T)=\{0\}$.

\item Prove that if Dim$_{F}(V)=$Dim$_{F}(W)=n\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, the $T$ is surjective if and only if Ker$(T)=\{0\}$

\item Prove that if Dim$_{F}(V)=$Dim$_{F}(W)\notin%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\cup\{0\}$, then neither direction of the previous result is always true.
\end{enumerate}

\item For $n\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ consider the vector space over $F$ given by%
\[
P_{n}(F)=\{f(x)\in F[x]\;|\;\text{Deg}(f)\leq n\}
\]


\begin{enumerate}
\item Prove that $\{1,(x-1),(x-1)^{2},(x-1)^{3}\cdots(x-1)^{n}\}$ is a basis
of $P_{n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$

\item Prove that $\{1,(x-1),(x-1)^{2},(x-1)^{3}\cdots(x-1)^{n}\}$ is a basis
of $P_{n}(%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
)$

\item Is $\{1,(x-1),(x-1)^{2},(x-1)^{3}\cdots(x-1)^{n}\cdots\}$ is a basis of
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack x]?$

\item Is $\{1,(x-1),(x-1)^{2},(x-1)^{3}\cdots(x-1)^{n}\cdots\}$ is a basis of
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack\lbrack x]]?$

\item Is $\{1,(x-1),(x-1)^{2},(x-1)^{3}\cdots(x-1)^{n}\}$ is a basis of
$P_{n}(%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{p})?$
\end{enumerate}

\item Let
\begin{align*}
a_{1,1}x_{1}+a_{1,2}x_{2}++a_{1,3}x_{3}+\cdots a_{1,2}x_{n}  &  =b_{1}\\
a_{2,1}x_{1}+a_{2,2}x_{2}++a_{2,3}x_{3}+\cdots a_{2,2}x_{n}  &  =b_{2}\\
a_{3,1}x_{1}+a_{3,2}x_{2}++a_{3,3}x_{2}+\cdots a_{3,n}x_{n}  &  =b_{3}\\
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots
\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots &
=\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\vdots\\
a_{m,1}x_{1}+a_{m,2}x_{2}++a_{m,3}x_{2}+\cdots a_{m,2}x_{n}  &  =b_{n}%
\end{align*}
be a system of linear equations over a field $F$. Explain how the set of
solutions to the system can be described as the kernel of a linear
transformation of vector spaces. (In the immortal words of Admiral Ackbar,
"It's a trap!")

\item Consider the real vector space $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack\lbrack x]]$.

\begin{enumerate}
\item Using what you know from calculus, argue that the formal derivative
$\frac{d}{dx}:$ $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack\lbrack x]]\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack\lbrack x]]$ is a linear transformation.

\item What are the Kernel and Image of this transformation.

\item Prove that for all $a\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, $a$ is an eigenvalue of $\frac{d}{dx}$.
\end{enumerate}

\item Let $W$ be a subspace of the vector space $V$ over the field $F$. Let
$T:V\rightarrow V$ be a linear transformation. We say $W$ is stable under $T$
if
\[
T(W)\subseteq W\text{.}%
\]


If $\lambda$ is an eigenvector of $T$, prove that both the eigenspace of
$\lambda$ and the generalized eigenspace of $\lambda$ are stable under $T$.

\item Let $V$ be a vector space over $F$. Let $T:V\rightarrow V$ be a linear
transformation. Can it happen that
\[
\text{Ker}(T)=\text{Im}(T)\text{?}%
\]

\end{enumerate}

\section{Lecture 2}

\begin{enumerate}
\item Consider the vector space over $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ given by%
\[
P_{n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)=\{f(x)\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\lbrack x]\;|\;\text{Deg}(f)\leq n\}
\]


Both $S_{1}=\{1,x,x^{2},\cdots x^{n}\}$ and $S_{2}=\{1,(x-1),(x-1)^{2}%
,(x-1)^{3}\cdots(x-1)^{n}\}$ are bases of $P_{n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$. Find the change of bases matrices
\[
\left(  _{S_{1}}\left\vert C\right\vert _{S_{2}}\right)  \text{ \ \ and
\ \ }\left(  _{S_{2}}\left\vert C\right\vert _{S_{1}}\right)
\]


\item Prove that any $2\times2$ real matrix with positive entries off the
diagonal is diagonalizable.

\item Let $M$ be any $n\times n$ matrix with rational entries. Let
$f(x)=\varkappa_{M}(x)$. \ Prove that if the greatest common divisor of $f(x)$
and its derivative $f^{\prime}(x)$ is $1$, then $M$ is diagonalizable over the
complex numbers.

\item Let
\[%
\begin{pmatrix}
a_{1,1} & a_{1,3} & a_{1,3} & \cdots & a_{1,n}\\
a_{2,1} & a_{2,2} & a_{2,3} & \cdots & a_{2,n}\\
a_{3,1} & a_{3,1} & a_{3,1} & \cdots & a_{3,n}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n}%
\end{pmatrix}
\]
be an $n\times n$ matrix over any field $F$. Prove that if for all
$k=1,2,\cdots(n-1)$,%
\[
\text{Det}\left(
\begin{pmatrix}
a_{1,k} & a_{1,k+1}\\
a_{2,k} & a_{2,k+1}%
\end{pmatrix}
\right)  =0
\]
Then
\[
\text{Det}\left(
\begin{pmatrix}
a_{1,1} & a_{1,3} & a_{1,3} & \cdots & a_{1,n}\\
a_{2,1} & a_{2,2} & a_{2,3} & \cdots & a_{2,n}\\
a_{3,1} & a_{3,1} & a_{3,1} & \cdots & a_{3,n}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n}%
\end{pmatrix}
\right)  =0\text{.}%
\]


\item A matrix of the form
\[
M=%
\begin{pmatrix}
1 & a_{1} & a_{1}^{2} & \cdots & a_{1}^{n-1}\\
1 & a_{2} & a_{2}^{2} & \cdots & a_{2}^{n-1}\\
1 & a_{n} & a_{3}^{2} & \cdots & a_{3}^{n-1}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
1 & a_{n} & a_{n}^{2} & \cdots & a_{n}^{n-1}%
\end{pmatrix}
\]
is called a Vandermonde matrix. Prove that its determinant is
\[%
%TCIMACRO{\tprod \limits_{1\leq i<j\leq n}}%
%BeginExpansion
{\textstyle\prod\limits_{1\leq i<j\leq n}}
%EndExpansion
(x_{i}-x_{j})
\]
by proving that for all $1\leq i<j\leq n$, must divide that determinant$.$

\item The Fibonacci sequence $F_{n}$ for $n=0,1,\cdots$ is given by%
\begin{align*}
F_{0}  &  =0\\
F_{1}  &  =1\\
F_{k+1}  &  =F_{k}+F_{k-1}%
\end{align*}


\begin{enumerate}
\item Find a $2\times2$ matrix $M$ so that
\[
M%
\begin{pmatrix}
F_{k-1} & F_{k}\\
F_{k} & F_{k+1}%
\end{pmatrix}
=%
\begin{pmatrix}
F_{k} & F_{k+1}\\
F_{k+1} & F_{k+2}%
\end{pmatrix}
\]


\item Find a nonrecursive formula for $F_{n}$.

\item Prove that $5$ always divides $F_{5k}$.
\end{enumerate}

\item Give necessary and sufficient conditions for a transformation
$T:V\rightarrow V$ on a finitely generated vector space over $F$ to have a
"square root$.$"

\item Let $M$ be the real matrix given by
\[
M=%
\begin{pmatrix}
0 & 0 & c\\
1 & 0 & -3\\
0 & 1 & 0
\end{pmatrix}
\]
Prove that if $|c|\leq2$, then $M$ is diagonalizable. (Hint: Expand
$2$Cos$(3\theta)$ in terms of $2$Cos$(\theta)$.)

\item Let $A$ and $B$ be square matrices over $F$, prove that $AB$ and $BA$
have the same characteristic polynomial.

\item Consider Mat$_{n\times n}(%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
)$ as the vector space $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{n^{2}}$. Prove that the set of diagonalizable matrices in Mat$_{n\times n}(%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
)$ is dense in $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{n^{2}}$, by proving

\begin{enumerate}
\item For all $A\in$ Mat$_{n\times n}(%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
)$ in Jordan canonical form and all real $\varepsilon>0$, \ there is a matrix
$B\in$ Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$ where all the entries are less than $\varepsilon$ and where $A+B$ is
diagonalizable$.$

\item Extend this to any $A\in$ Mat$_{n\times n}(%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
)$ no matter the form$.$
\end{enumerate}

\item Let $T:V\rightarrow V$ be a linear transformation on a vector space $V$
over $F$ . We say $T$ is nilpotent, if there is $k\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ so that $T^{k}=0$.

\begin{enumerate}
\item Prove that if $V$ has dimension $n$ then, if $T$ is nilpotent then
$T^{n}=0$.

\item Let Dim$_{F}(V)=5$. For each $k=1,2,3,4,5$, give an example of a
nilpotent transformation $T:V\rightarrow V$ so that $k$ is the smallest
exponent with $T^{k}=0$.
\end{enumerate}

\item Let
\[
M=%
\begin{pmatrix}
-56 & 237 & 37 & 2031\\
-139 & 595 & 98 & 5229\\
-669 & 2874 & 468 & 25129\\
27 & -116 & -19 & -1017
\end{pmatrix}
\]


Find the Jordan Canonical Form of $M$.

\item Find all the matrices over $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{3}$ with minimum polynomial $(x-1)(x-2)$ and characteristic polynomial
$(x-1)(x-2)^{2}$.

\item Find all the matrices over $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{3}$ with minimum polynomial $(x-1)(x-2)^{2}$ and characteristic polynomial
$(x-1)(x-2)^{2}$.

\item Let $M\in$ Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$

\begin{enumerate}
\item Prove that $M$ also acts on Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$ \ as a vector space of dimension $n^{2}.$ Let $T:$Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)\rightarrow$Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)$ be the transformation $T(A)=MA$.

\item Suppose that $M$ has distinct eigenvalues $\{\lambda_{1}\cdots
\lambda_{n}\}$. Prove that $T$ is diagonalizable as an $n^{2}\times n^{2}$ matrix.
\end{enumerate}

\item Let $M=(a_{i,j})\in$ Mat$_{n\times n}(F)$. The trace of $F$ is%
\[
\text{Tr}(M)=%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
a_{k,k}%
\]


\begin{enumerate}
\item Prove that if $\varkappa_{M}(x)=%
%TCIMACRO{\tsum \limits_{k=0}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=0}^{n}}
%EndExpansion
b_{k}x^{k}$, then Tr$(M)=b_{n-1}$.

\item Prove that if $A$ is an $n\times n$ matrix with an inverse, then%
\[
\text{Tr}(AMA^{-1})=\text{Tr}(M)
\]


\item Prove that Tr$(M)$ is the sum of the eigenvalues of $M$ with
multiplicities$.$

\item Prove that for any $n\times n$ matrices $A$ and $B$, Tr$(AB)=$Tr$(BA)$
\end{enumerate}
\end{enumerate}

\section{Lecture 3}

\begin{enumerate}
\item Consider the vector space $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$.

\begin{enumerate}
\item Prove that the dot product makes $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$ an inner product space in a standard way$.$

\item Find an orthonormal basis for this space.

\item If $n=3$ and $W=<v_{1},v_{2}>$ where $v_{1}$ and $v_{2}$ are linearly
independent, then $W^{\bot}=<v_{1}\times v_{2}>.$

\item What would you say makes $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{n}$ an inner product space in a standard way?
\end{enumerate}

\item Let $V$ be a inner product space. Let $T:V\rightarrow V$ be an
automorphism. Explain why the definition of self adjoint,
\[
<T(v),w.>=<v,T(w)>
\]
allows us to say $T$ "preserves the inner product." and thus consider $T$ as a
transformation of inner product spaces.

\item Give a matrix of a symmetric bilinear form on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$ using the standard basis with no zero entries and with signature
$[2,1,1]$.

\item Let $V$ be a finitely generated vector space over the field $F$. Let $B$
be a nondegenerate bilinear form on $V$. Prove that for any subspace $W$ of
$V$, Dim$_{F}(W)+$Dim$_{F}(W^{\bot}=$Dim$_{F}(V))$.

\item Prove if $B\ $is a bilinear form on the vector space $V$, then $B$ is
reflexive if and only if it is either alternating or symmetric.

\item Explain why the eigenvectors of a self adjoint matrix must be real.

\item Consider the vector space $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$.

\begin{enumerate}
\item If possible, give an example of a non-trivial symmetric bilinear form on
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$ with a non-zero isotropic vector.

\item If possible, give an example of a non-trivial symmetric bilinear form on
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$ with a non-zero isotropic vector and an anisotropic vector

\item If possible, give an example of a non-trivial symmetric bilinear form on
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$ with a nontrivial isotropic subspace

\item If possible, give an example of a non-trivial symmetric bilinear form on
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$ with a nontrivial coisotropic subspace

\item If possible, give an example of a non-trivial symmetric bilinear form on
$%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{3}$ with a nontrivial Lagrangian subspace.
\end{enumerate}

\item Consider elements of $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}$ to be column vectors. Let
\[
M=%
\begin{pmatrix}
1 & 1 & 2 & 0\\
1 & 3 & 4 & -1\\
2 & 4 & 6 & -1\\
0 & -1 & -1 & 0
\end{pmatrix}
\]


Let $B:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}\times%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be given by
\[
B(v,w)=v^{T}Mw\text{.}%
\]


\begin{enumerate}
\item Prove that $B$ is a symmetric bilinear form on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}$.

\item If possible, find a non-zero isotropic vector.

\item If possible,find an anisotropic vector

\item If possible,find a nontrivial isotropic subspace

\item If possible, find a nontrivial coisotropic subspace

\item If possible, nontrivial Lagrangian subspace.

\item Compute the associated quadratic form with respect to the basis
$\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0.0.0.1)\}$

\item Compute the associated quadratic form with respect to the basis
$\{(1,0,0,0),(1,1,0,0),(1,1,1,0),(1.1.1.1)\}.$
\end{enumerate}

\item Let
\[
M=%
\begin{pmatrix}
a & b\\
b & c
\end{pmatrix}
\]
where $a,b,c$ $b\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $ac-b^{2}\neq0$. Let $B:%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}\times%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}\rightarrow%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ be given by
\[
B(v,w)=v^{T}Mw\text{.}%
\]


\begin{enumerate}
\item Prove that $B$ is a nondegenerate Hermitian form on $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}$.

\item Find an orthonormal basis of $%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}$ with respect to the inner product given by $B$.
\end{enumerate}

\item Let
\[
M=%
\begin{pmatrix}
a & b\\
b & c
\end{pmatrix}
\]
where $a,b,c$ $b\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $ac-b^{2}\neq0$. \ \ Let $T:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ be given by $T(v)=Mv$ and consider $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ as an inner product space under the dot product.

\begin{enumerate}
\item Prove $<T(v),w>=<v,T(w)>$

\item Find an orthogonal basis of eigenvalues of $T$.
\end{enumerate}

\item Let $V$ be an inner product space, and let $S=\{v_{1},\cdots v_{n}\}$ be
an orthonormal basis of $V$. \ For $w=%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
a_{i}v_{i}$ , compute $||w||$.

\item Consider Mat$_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
).$

\begin{enumerate}
\item Prove that
\[
B:\text{Mat}_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)\times\text{Mat}_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)\rightarrow\text{Mat}_{n\times n}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
)
\]
given by $B(N,M)=$Tr$(AB)$ is a symmetric bilinear form.

\item Determine the signature of this real symmetric bilinear form.
\end{enumerate}

\item Let $M$ be a symmetric $n\times n$ matrix over $%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
$. Prove that all the eigenvalues of $M$ are real.

\item Consider  $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$ as an innerproduct space under the dot product 

\begin{enumerate}
\item For $n=2$, give an example of an othogonal basis of $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$ that is not the standard basis $\{(1,0),(0,1)\}$

\item For $n=3$, give an example of an othogonal basis of $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$\ that is not the standard basis $\{(1,0,0),(0,1,0),(0,0,1)\}$

\item For $n=4$, give an example of an othogonal basis of $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$\ that is not the standard basis $\{(1,0,0),(0,1,0),(0,0,1)\}$
\end{enumerate}

\item The set of functions $f:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ that are continuous on $[0,1]$ form a vector space over $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$. \ Call it $V$

\begin{enumerate}
Prove that $<f_{1},f_{2}>=\int_{0}^{1}f_{1}(x)f_{2}(x)\,dx$ is a nondegenerate
symmetric bilinear form on $V$
\end{enumerate}


\end{enumerate}


\end{document}