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\usepackage{pandora,euler,amsmath,amsfonts}

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

\begin{document}

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

\maketitle

\begin{enumerate}

\item Let
\begin{equation*} A =
\begin{pmatrix}
   1 & -1 &  1 &  0 \\
   1 & -1 &  2 & -1 \\
   0 &  0 &  7 & -7 \\
   5 & -5 &  1 &  4
\end{pmatrix}, \qquad
\boldsymbol{b} = 
\begin{pmatrix}
  2 \\ 3 \\ 7 \\ 6
\end{pmatrix}.
\end{equation*}
Find:
\begin{enumerate}

\item The solution set of the equation $A \boldsymbol{x} =
\boldsymbol{b}$. (Do not use back-substitution.  Use the method taught
in class.)

\item A basis for the row space of $A$.

\item A basis for the null space of $A$.

\item A basis for the column space of $A$.

\item An example of a vector $\boldsymbol{c}$ such that the equation
$A \boldsymbol{x} = \boldsymbol{c}$ does not have a solution.

\item $\rank A$ and $\nullity A$; verify that for this matrix the
``rank equation'' holds.

\item Extend the basis found in part (d) to a basis of $\mathbb{R}^4$.

\item Is $A$ invertible?  Give at least \emph{four} different reasons
for why or why not.

\end{enumerate}
\hfill (10+5+5+5+5+5+5+5+5)

\item Let $V$ be a  vector space with basis $\{v_1, v_2, v_3\}$.
Prove that $\{v_1, v_1 - v_2, v_1 - v_2 - v_3\}$ is also a basis for
$V$.
\hfill (10)

\vspace*{\fill}
\item Prove that, if $W$ is a subspace of an $n$-dimensional vector
space $V$ and $\dim W = n$, then $W=V$.
\hfill (10)

\vspace*{\fill}

\item Let $W$ be a subspace of $\mathbb{R}^n$ and $T \colon
\mathbb{R}^n \to \mathbb{R}^n$ a linear transformation.  Prove that
\begin{equation*}
  U = T(W) = \{ T(w) \colon w \in W \}
\end{equation*}
is a subspace of $\mathbb{R}^n$.
\hfill (10)

\item Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear
transformation.
\begin{enumerate}
\item Show that $T(0) = 0$.

\item Is $S \colon \mathbb{R}^n \to \mathbb{R}^n$, defined through
\begin{equation*}
  S(\boldsymbol{x}) = 3 \, \boldsymbol{x} +
    \begin{pmatrix}
      1 \\ 1 \\ \vdots \\ 1
    \end{pmatrix},
\end{equation*}
a linear transformation?

\item Give an example of two linear transformations $S$ and $T$ where
$S \circ T \neq T \circ S$.

\end{enumerate}
\hfill (5+5+10)
  
\end{enumerate}

\end{document}
