\documentclass[11pt]{article}

\usepackage{pandora,euler,amsmath,amsfonts}

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

\title{Math 3A, Section B, Spring 1997}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle

\begin{enumerate}

\item Solve the system of linear equations
\begin{equation*}
\begin{pmatrix}
   -1 & -4 & -4 & -9 \\   % -1 & -5 & -4 & -10 \\ 
   -2 & -8 & 8 & 30 \\    % -2 & -10 & 8 & 28  \\
   -3 & -12 & -1 & 6 \\   % -3 & -15 & -1 & 3  \\
   -4 & -16 & -2 & 6      % -4 & -20 & -2 & 2
\end{pmatrix}
\begin{pmatrix}
  x_1 \\ x_2 \\ x_3 \\ x_4
\end{pmatrix} =
\begin{pmatrix}
  9 \\ -14 \\ 5 \\ 8      % 2 \\ 4 \\ 6 \\ 8
\end{pmatrix}.
\end{equation*}
Use the method taught in class.  Check your answer (required!).
\hfill (15)

\item Let $V = P_3$, the vector space of polynomials of degree less or
equal to 3.  Find a basis for $V$ containing the vectors
\begin{align*}
  \boldsymbol{v}_1 & = 1-2\,x, \\
  \boldsymbol{v}_2 & = 1+ x + x^2 + x^3, \\
  \boldsymbol{v}_3 & = 1-x^2-x^3.
\end{align*}
\hfill(10)

\item
\begin{enumerate}

\item Diagonalize the matrix
\begin{equation*}
\begin{pmatrix}
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  2 & 2 & 3     % 3 & 3 & 4 
\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 Find $A^{10}$.
\end{enumerate}
\hfill (10+5)


\item
\begin{enumerate}
\item Find an orthonormal basis for 
\begin{equation*}
  W = \Span \left\{
    \begin{pmatrix}
      -1 \\ 1 \\ 0    % 1 \\ -1 \\ 0
    \end{pmatrix},
    \begin{pmatrix}
      2 \\ 1 \\ 6     % 1 \\ 2 \\ 6
    \end{pmatrix} \right\} .
\end{equation*}
\item Find the projection of $\displaystyle v = \begin{pmatrix} 3 \\ 2
\\ 1 \end{pmatrix}$ onto $W$.
\end{enumerate}
\hfill (5+5)


\item Let $T \colon \mathbb{R}^2 \to \mathbb{R}^4$ be defined by
\begin{equation*}
  T (\boldsymbol{x}) =
    \begin{pmatrix}
      x_1 \\ x_1 + x_2 \\ x_1 + 2 x_2 \\ x_1 + 3 x_2
    \end{pmatrix}.
\end{equation*}
Find the matrix $T_{B,E}$ which represents $T$ when $\mathbb{R}^2$ is
endowed with the basis
\begin{equation*}
  B = \left\{
    \begin{pmatrix}
      1 \\ 2
    \end{pmatrix},
    \begin{pmatrix}
      0 \\ -1
    \end{pmatrix} \right\}
\end{equation*}
and $\mathbb{R}^4$ is endowed with the standard basis
$E=\{\boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3,
\boldsymbol{e}_4\}$.
\hfill (10)

\item
\begin{enumerate}

\item Is the set of all matrices for which the sum of the diagonal
elements is 0, endowed with the usual addition and scalar
multiplication, a vector space?  Explain.

\item Is the set of all matrices for which the sum of the diagonal
elements is 1, endowed with the usual addition and scalar
multiplication, a vector space?  Explain why this is different from
the previous case.
\end{enumerate}
\hfill (5+5)

\item A matrix $A \in M(n \times n)$ is called \emph{nilpotent} if
there exists a positive integer $k$ such that $A^k = 0$.  Prove that a
nilpotent matrix is not invertible.
\hfill(5)

\item Let $V$ be the vector space of continuous functions on the
interval $[0, 2\pi]$.  Recall from Calculus that the mean (or average)
of a function is defined
\begin{equation*}
  M(f) = \frac1{2\pi} \int_0^{2\pi} f(x) \, dx
\end{equation*}
for every $f \in V$.
\begin{enumerate}

\item Prove that $M$ is a linear transformation on $V$.

\item Let $W$ be a subspace of $V$ with basis
\begin{equation*}
  B = \{ 1, \sin x, \cos x, \sin 2x, \cos 2x \}
\end{equation*}
and consider $M$ restricted to $W$.  Then
\begin{gather*}
  \dim W = \fbox{\vphantom{T}\hphantom{888}}, \\
  \dim \range M = \fbox{\vphantom{T}\hphantom{888}}, \\
  \dim \ker M = \fbox{\vphantom{T}\hphantom{888}}.
\end{gather*}
Fill in the blanks and explain \emph{briefly} how you got your answer.

\item Describe $\ker M$ as concretely as you can.
\end{enumerate}
\hfill (5+5+5)

\item Let $A \in M(n \times n)$ satisfy the equation
\begin{equation*}
  AA^T = I.
\end{equation*}
(Such a matrix is called \emph{orthogonal}.)
\begin{enumerate}

\item Is $A$ invertible? Explain.

\item Prove that if $\lambda$ is an eigenvalue of $A$, then either
$\lambda = 1$ or $\lambda = -1$.

Hint:  Compare the eigenvalues of $A$ with those of $A^{-1}$.
\end{enumerate}
\hfill (5+5)

  
\end{enumerate}

\end{document}
