\documentclass[11pt]{article}

\usepackage{pandora,euler,amsmath,amsfonts}

\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{Midterm Exam}

\maketitle


\begin{enumerate}

\item Let
\begin{equation*} A =
\begin{pmatrix}
   0 &  0 &  1 &  6 \\
   1 & 10 &  1 &  1 \\
  -1 &-10 &  0 &  5 \\
   2 & 20 &  3 &  6
\end{pmatrix}, \qquad
\boldsymbol{b} = 
\begin{pmatrix}
 -4 \\ 1 \\-5 \\ -4
\end{pmatrix}.
\end{equation*}
\begin{enumerate}

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

\item Find a basis for $\ker (A)$ and $\range (A)$. (In other words,
find a basis for the null space and the column space of $A$.)

\end{enumerate}
\hfill (30)


\item Let $V$ be a vector space, and $\{ v_1, \ldots, v_4 \} \subset
V$ a set of linearly independent vectors (so $\dim V \geq 4$).  Let
\begin{alignat*}{4}
  w_1 &= v_1  + && v_2 + &&v_3 + && v_4 \\
  w_2 &= v_1  + & 2\, & v_2 + & 3 \,& v_3 + & 3 \, &v_4 \\
  w_3 &=  2\, v_1   & + 2 \,& v_2 + & 2\,& v_3 + & 6 \,& v_4 \\
  w_4 &=  2\, v_1   & + 3 \,& v_2 + & 4\,& v_3 + & 7 \,& v_4  
\end{alignat*}
\begin{enumerate}
\item Use coordinate vectors to show that $w_1, \ldots, w_4$ are
linearly dependent.
\item Find a basis for $\Span \{ w_1, \ldots, w_4 \}$.
\end{enumerate}
\hfill (10+5)


\item Let $T \colon \mathbb{R}^3 \to \mathbb{R}^3$ be defined by
\[
  T(\boldsymbol{x}) =
  \begin{pmatrix}
  x_3 \\ x_2 + x_3 \\ x_1 + x_2 + x_3
  \end{pmatrix}
\]
\begin{enumerate}
\item Find the standard matrix representation of $T$.
\item Is $T$ invertible?  If yes, find the standard matrix
representation for $T^{-1}$.
\end{enumerate}
\hfill (5+10)

\item Let $V$ be a vector space with basis $\{v_1,\dots, v_n\}$, and
consider an arbitrary nonzero vector $w \in V$.
\begin{enumerate}
\item Show, by explicitly using the definition of linear independence,
that the set $\{v_1,\dots, v_n, w\}$ is linearly dependent.
\item Show that there exists some index $i \in \{1,\dots, n\}$ such
that the set
\[
  \{ v_1,\dots, v_{i-1}, w, v_{i+1}, \dots, v_n\}
\]
is a basis of $V$.  (In other words, prove that you can always obtain
another basis for $V$ by replacing one of the basis vectors by $w$.)
\item Extend the argument from part (b) to show that any set of $n$
linearly independent vectors in $V$ is a basis.
\end{enumerate}
\hfill (5+10+5)


\item Let $\boldsymbol{v} \in \mathbb{R}^n$ be fixed.  Consider the
transformation $S \colon \mathbb{R}^n \to \mathbb{R}$ defined through
\[
  T(\boldsymbol{x}) = \boldsymbol{v} \cdot \boldsymbol{x}.
\]
\begin{enumerate}
\item Show that $T$ is a linear transformation.
\item Find the standard matrix representation of $T$.
\item Describe $\ker T$ geometrically.
\end{enumerate}
\hfill(10+5+5)
\end{enumerate}


\end{document}
