\documentclass[11pt]{article}

\usepackage{pandora,euler,amsmath,amsfonts}

\newcommand{\Span}{\operatorname{span}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\nullity}{\operatorname{nullity}}

\begin{document}
\begin{center}
{\LARGE {Mathematics 3A, Section A, Winter 1997}}
\end{center}
\begin{center}
\large Final Exam
\end{center}

\begin{enumerate}

\item Solve the system of linear equations
\begin{equation*}
\begin{pmatrix}
   1 & -1 & -3 & -3 \\
   2 &  1 &  9 & 12 \\
   3 &  1 & 11 & 15 \\
   0 & -1 & -5 & -6
\end{pmatrix}
\begin{pmatrix}
  x_1 \\ x_2 \\ x_3 \\ x_4
\end{pmatrix} =
\begin{pmatrix}
  -3 \\ 15\\ 19\\ -7
\end{pmatrix}.
\end{equation*}
Use the method taught in class.  Check your answer in a meaningful way
(required!).

\item Let $V$ be a the vector space of continuous real functions.
Find a basis for the subspace
\begin{equation*}
  W = \Span \{ 1 + \cos x, -1 + \cos 2x,
               \cos x + \cos 2x, 1 + \cos 3x \}  
\end{equation*}

\item Compute the determinant
\begin{equation*}
\begin{vmatrix}
  1 & 3 & 0 & 0 & 0 & 0 \\
  2 & -1& 0 & 0 & 0 & 0 \\
  0 & 0 & -1& 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 0 & 0 & 0 & 5 & 0 \\
  0 & 0 & 0 & 4 & 0 & 0 
\end{vmatrix}
\end{equation*}

\item Diagonalize the matrix
\begin{equation*}
\begin{pmatrix}
  -2& 0 & -3 \\
  0 & 1 & 0  \\
  -3& 0 & -2 
\end{pmatrix}.
\end{equation*}
I.e., find the eigenvalues $\lambda_i$ of $A$, an invertible matrix
$C$, and a diagonal matrix $D$ such that $D=C^{-1} A C$.

\item
\begin{enumerate}
\item Find an orthonormal basis for 
\begin{equation*}
  W = \Span \left\{
    \begin{pmatrix}
      1 \\ 1 \\ 1
    \end{pmatrix},
    \begin{pmatrix}
      2 \\ -1 \\ 2
    \end{pmatrix} \right\} .
\end{equation*}
\item Find the projection of $\displaystyle v = \begin{pmatrix} 1 \\ 2
\\ 3 \end{pmatrix}$ onto $W$.
\end{enumerate}

\item Let $T \colon \mathbb{R}^3 \to \mathbb{R}^2$ be defined by
\begin{equation*}
  T (\boldsymbol{x}) =
    \begin{pmatrix}
      x_1 + x_2 + x_3 \\ -x_2
    \end{pmatrix}.
\end{equation*}
Find the matrix $T_{B,B'}$ which represents $T$ when $\mathbb{R}^3$ is
endowed with the basis
\begin{equation*}
  B = \left\{
    \begin{pmatrix}
      1 \\ 0 \\ 1
    \end{pmatrix},
    \begin{pmatrix}
      0 \\ 1 \\ 1
    \end{pmatrix},
    \begin{pmatrix}
      1 \\ 1 \\ 0
    \end{pmatrix} \right\}
\end{equation*}
and $\mathbb{R}^2$ is endowed with the basis
\begin{equation*}
  B' = \left\{
    \begin{pmatrix}
      1 \\ 1
    \end{pmatrix},
    \begin{pmatrix}
      0 \\ -1
    \end{pmatrix} \right\}.
\end{equation*}

\item Let $W \subset \mathbb{R}^n$ be a subspace.  Recall that the
orthogonal complement of $W$ is defined
\begin{equation*}
  W^\bot = \{ v \in \mathbb{R}^n \colon
    v \text{ is orthogonal to every vector of } W \}.
\end{equation*}
\begin{enumerate}
\item Prove that $W^\bot$ is a subspace of $\mathbb{R}^n$.
\item Let $\{ w_1, \dots, w_k\}$ be a basis of $W$ and $\{ u_1,
\dots, u_m\}$ be a basis of $W^\bot$.  We have shown in class that
every vector $v \in \mathbb{R}^n$ can be expressed uniquely in the
form $v = w + w^\bot$ for $w \in W$ and $w^\bot \in W^\bot$.  Use this
fact to prove that
\begin{equation}
  \{ w_1, \dots, w_k, u_1, \dots, u_m \}
\end{equation}
is a basis of $\mathbb{R}^n$.
\end{enumerate}

\item Let $A,B \in M(n \times n)$.  Prove, by using determinants or
otherwise, that if $AB$ is invertible, then $A$ and $B$ are both
invertible.

\item Let $A \in M(n \times n)$ and $\lambda \in \mathbb{R}$.  Prove
that the set $E_\lambda = \{ v \in \mathbb{R}^n \colon A v = \lambda v
\}$ is a subspace of $\mathbb{R}^n$.

\item Let $A \in M(n \times n)$ be a \emph{symmetric} matrix.  Let
$v_1$ and $v_2$ be eigenvectors of $A$ with eigenvalues $\lambda_1$
and $\lambda_2$.  Prove that if $\lambda_1 \neq \lambda_2$, then $v_1
\bot v_2$. 
\end{enumerate}

\end{document}
