\documentclass[11pt]{article}

\usepackage{pandora,euler,amsmath,amsfonts,fullpage}

\newcommand{\Span}{\operatorname{span}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\nullity}{\operatorname{nullity}}
%\newcommand{\ker}{\operatorname{ker}}
\newcommand{\range}{\operatorname{range}}

\begin{document}

\title{Math 3A, Section B, Spring 1998}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle


\begin{enumerate}

\item Solve the system of linear equations
\begin{equation*}
\begin{pmatrix}
    1 &  2 &  6 &  9 \\   
   -2 &  1 & -2 & -3 \\    
   -1 &  2 &  2 & 3 \\   
   -1 &  1 &  0 & 0      
\end{pmatrix}
\begin{pmatrix}
  x_1 \\ x_2 \\ x_3 \\ x_4
\end{pmatrix} =
\begin{pmatrix}
  12 \\ -4 \\ 4 \\ 0      
\end{pmatrix}.
\end{equation*}
Use the method taught in class.  Check your answer (required!).
\hfill (15)



\item Compute the determinant
\begin{equation*}
\begin{vmatrix}
  0 & 1 & 0 & 1 \\
  1 & 0 & 1 & 0 \\
  0 & 1 & 0 & -1 \\
  1 & 0 & -1& 0 
\end{vmatrix}
\end{equation*}
\hfill (5)


\item Let 
\begin{equation*} A = 
\begin{pmatrix}
   3 & 4 & 0 \\  
   4 & -3 & 0 \\   
   0 & 0 & 3    
\end{pmatrix}.
\end{equation*}

\begin{enumerate}
\item Diagonalize $A$.  I.e., find the eigenvalues $\lambda_i$ of $A$,
an invertible matrix $C$, and a diagonal matrix $D$ such that $A = C D
C^{-1}$.

\item Let $p(\lambda)$ be the characteristic polynomial of $A$.  If
you multiply out the characteristic polynomial you used in part (a),
you should find
\begin{equation*}
  p(\lambda) = -\lambda^3 + 3 \, \lambda^2 + 25 \, \lambda^1
    - 75 \, \lambda^0.
\end{equation*}
Show that $p(A)=0$ (the zero matrix).

Hints:
\begin{itemize}
\item Note that $A^0=I$.
\item It is easiest to use the diagonalization from part (a), although
you can of course do the computation with $A$ directly.
\end{itemize}
\hfill (10+5)

\end{enumerate}


\item Consider the vector space of exponential functions
\begin{equation*}
  V = \Span \{ e^x, e^{-x} \},
\end{equation*}
with standard basis $E = \{ e^x, e^{-x} \}$.
\begin{enumerate}
\item Find the standard matrix representation of the derivative
transformation $T\colon V \to V$,
\begin{equation*}
  T(v) = \frac{dv}{dx}.
\end{equation*}

\item Is $T$ invertible? If yes, what is the inverse?

\item Show that $B = \{ 2e^x - e^{-x}, e^x - e^{-x} \}$ is a basis for
$V$.

\item Find the matrix representation $T_B$ ($=T_{B,B}$) of $T$ with
respect to the basis $B$.

\item Find the coordinates of $v= -3 \, e^x + 2 \, e^{-x}$ with
respect to the basis $B$.

\item What are the eigenvectors and eigenvalues of $T$?
\end{enumerate}
\hfill (5+5+5+5+5+5)



\item Gram--Schmidt orthonormalization: Recall that, given three
linearly independent vectors $\{u_1, u_2, u_3\}$, one finds an
orthonormal basis $\{w_1, w_2, w_3\}$ for their span by the following 
procedure:
\begin{align*}
  w_1 & \equiv \frac{u_1}{\| u_1 \|}, \\
  v_2 & \equiv u_2 - (u_2 \cdot w_1) w_1, &
  w_2 & \equiv \frac{v_2}{\| v_2 \|}, \\
  v_3 & \equiv u_3 - (u_3 \cdot w_1) w_1 - (u_3 \cdot w_2) w_2, &
  w_3 & \equiv \frac{v_3}{\| v_3 \|}.
\end{align*}
\begin{enumerate}
\item Show that the vectors $\{ w_1, w_2, w_3 \}$ are pairwise
perpendicular.

\item Apply the Gram--Schmidt procedure to find an orthonormal basis
for
\begin{equation*}
  \Span \left\{
    \begin{pmatrix}
      1 \\ 1 \\ 1 \\ 1    
    \end{pmatrix},
    \begin{pmatrix}
      1 \\ 0 \\ 0 \\ 1
    \end{pmatrix} \right\} .
\end{equation*}
\end{enumerate}
\hfill (5+5)


\item Prove that $A$ and $A^T$ have the same eigenvalues.

Hint: Consider the characteristic polynomials.
\hfill (5)

\item 
\begin{enumerate}
\item Prove that $\range AC \subset \range A$ for all matrices $A$ and
$C$ such that their product is defined.

\item Use the statement from part (a) to show that $\rank AC \leq
\rank A$.
\end{enumerate}
\hfill (5+5)

\item Prove that $A \in M(n \times n)$ is invertible if and only if no
eigenvalue is zero.
\hfill (5+5)

\end{enumerate}



\end{document}
