\documentclass{article}
\usepackage{amsmath,amsthm,amssymb,amsfonts,fancyhdr,graphicx,multicol}
\usepackage[margin=1.5in]{geometry}


\renewcommand{\baselinestretch}{1.1}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem{remark}{Remark}
\newtheorem{definition}{Definition}
\newcommand{\bp}{\begin{problem}}
\newcommand{\ep}{\end{problem}}

\newtheorem*{answer}{Answer}
\newtheorem*{lemma*}{Lemma}
\newtheorem*{theorem*}{Theorem}
\newcommand{\ba}{\begin{answer}}
\newcommand{\ea}{\end{answer}}

\newcommand{\ds}{\displaystyle}
\newcommand{\ben}{\renewcommand{\theenumi}{\alph{enumi}}\renewcommand{\labelenumi}{(\theenumi)}\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

\newcommand{\Z}{\mathbb{Z}} 
\newcommand{\N}{\mathbb{N}} 
\newcommand{\R}{\mathbb{R}} 
\newcommand{\C}{\mathbb{C}} 
\newcommand{\Rep}{\operatorname{Rep}}
\newcommand{\id}{\operatorname{Id}}
\newcommand{\tr}{\operatorname{trace}}

\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}


\fancyhead[L]{\Large \textbf{Math 207 Fall 2014}}
\fancyhead[R]{\large \textbf{Practice Exam}}


\begin{document}
\pagestyle{fancy}
\fancyfoot[R,L,C]{}


\bp Let $V$ be the vector space of all $n\times n$ complex matrices.
Prove that $\<A,B\>=\tr(AB^*)$ defines an inner product on $V$. 
\ep

\bp Let $A$ be any $n\times n$ complex matrix.  Prove that $A$ and
$A^t$ are similar.
\ep

\bp Prove that if $A^2=A$ then $A$ is diagonalizable.
\ep


\bp Let $V$ be the space of real polynomials of degree $\leq 3$ and
let $\sigma:V \to V$ be the differentiation operator 
$$\sigma(p(t))=\frac{dp(t)}{dt}.$$
\ben
\item What is the adjoint of $\sigma$ if we equip $V$ with the 
inner product defined by $$\<p,q\>:=\int_0^1 p(t)q(t)dt?$$
\item What is the adjoint of $\sigma$ if we equip $V$ with the 
inner product defined by $$\<p,q\>:=p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1)?$$
\een
\ep

\bp Let $M=\left(
\begin{array}{ccc}
 15 & -42 & 12 \\
 -3 & 10 & -6 \\
 -10 & 30 & -11 \\
\end{array}
\right)$.  Find a matrix $N$ so that $N^2=M$.
\ep


\bp Suppose that $A$ is a $2\times 2$ real matrix and let
$v\in \R^2$ be any vector.  Since 
$\R^2$ is two dimensional, the three vectors $\{v,Av,A^2v\}$ must be
linearly dependent.  Prove that, in fact, $A^2v=\alpha Av+\beta v$
where $\alpha=\tr(A)$ and $\beta=-\det(A).$
\ep


\bp Suppose $V$ is an $n$ dimensional vector space.
An operator $T:V\to V$ is called nilpotent if $T^k=0$
for some integer $k$. 
\ben
\item Prove that if $T$ is nilpotent,
  then $T^n=0$.
\item Prove that if $T$ is nilpotent, then there is a basis of
  vectors of $V$ so that the matrix for $T$ is upper triangular
  with zeroes on the diagonal.
\een
\ep


\bp Let $V$ be the space of polynomials of degree less than or equal
to $2$.  Let $T:V \to V$ be the homomorphism defined by 
$$p(x)\overset{T}{\mapsto}p(1)x^2+p'(0)x+p''(0)$$
\ben
\item Compute the determinant and trace of $T$.
\item Use the ordered basis $B=\{x^2,x,1\}$ to express $T$ as a
  $3\times 3$ matrix $A=\Rep_{B,B}(T)$.
\item Find a basis $E$ consisting of eigenvectors for $T$.
\item Let $Q$ be the change of basis matrix $Q=\Rep_{B,E}(\id)$.
\item Use your the ordered basis $E$ to express $T$ as a
  $3\times 3$ matrix $B=\Rep_{E,E}(\id)$.
\een
\ep

\bp Let $$C=
\left(
\begin{array}{ccccc}
 3 & 3 & 2 & 0 & 1 \\
 3 & -5 & -2 & -4 & 1 \\
 6 & 6 & 4 & 0 & 2 \\
 3 & 5 & 3 & 1 & 1 \\
\end{array}
\right)
$$
\ben
\item Find a basis for the kernel of $C$.
\item Find a basis for the image of $C$.
\item Find a basis for the space spanned by the columns of $C$.
\item Find a basis for the space spanned by the columns of $C^t$.
\een
\ep


\bp Let 
$$A=
\left(
\begin{array}{ccc}
 -1 & 8 & 2 \\
 -1 & -7 & -1 \\
 2 & 8 & -1 \\
\end{array}
\right)
\text{ and }
B=\left(
\begin{array}{ccc}
 7 & 1 & 3 \\
 7 & 3 & 4 \\
 -10 & -2 & -4 \\
\end{array}
\right)
$$
\ben
\item Find the characteristic and minimum polynomials and the Jordon
  canonical forms for $A$ and $B$.
\item Find matrices $P$ and $Q$ so that $P^{-1}AP$ and $Q^{-1}BQ$ are in
Jordon form.  
\item Compute $e^A$ and $e^B$.
\een
\ep

\bp Solve the two second order differential equations:
\begin{align*}
y_2''&=y_2-2y'_1\\
y_1''&=y_1+2y_2'
\end{align*}
subject to the initial conditions $$y_1(0)=0,\quad y_2(0)=0, \quad
y_1'(0)=0, \quad y_2'(0)=1.$$
By translating them into a single system of the form $Y'(t)=A Y(t)$
for a $4\times 4$ matrix $A$.
\ep


\end{document}

