\documentclass[11pt]{amsart}
% PREDOC SETTINGS
  \pagestyle{empty}
  \addtolength{\hoffset}{-48pt}
  \addtolength{\textwidth}{96pt}
  \addtolength{\voffset}{-32pt}
  \addtolength{\textheight}{45pt}
  \usepackage{graphicx}
  %latexsym,amsmath,amssymb,amsfonts,MnSymbol,epsfig}
  %\usepackage[all]{xy}
  \newcommand{\RR}{{\mathbb R}}
  \newcommand{\bi}{\begin{itemize}}
  \newcommand{\ei}{\end{itemize}}
  \newcommand\ben{\begin{enumerate}}
  \newcommand\een{\end{enumerate}}
  \newcommand{\be}{\begin{equation}}
  \newcommand{\ee}{\end{equation}}
  \newcommand{\bea}{\begin{eqnarray}}
  \newcommand{\eea}{\end{eqnarray}}
  \newcommand\vg{\vspace{2ex}}
  \newcommand{\bc}{\begin{center}}
  \newcommand{\ec}{\end{center}}
  \newcommand{\ie}{{\it i.e.\ }}
  \newcommand{\eg}{{\it e.g.\ }}
  \newcommand{\mbf}[1]{{\mathbf #1}}
% DOCUMENT
\begin{document}
  % INTRO
    \noindent
    Your name:
    \vskip 12pt
    \noindent
    Instructor (please circle):
    \hspace{1in}
    \noindent
    Alex Barnett
    \hskip .7in
    Michael Musty
    \vskip 12pt
    \noindent
        {\bf \large Math 22 Summer 2017, Homework 1, due Fri June 30}
        \quad
    {\em Please show your work, and check your answers. No credit is given for solutions without work or justification.}
    \vskip .25in
  % NUMBERED QUESTIONS
  \ben
    \item
      % Question 1
      Given the system of equations, answer the following questions.
      \bea
        2x_1+4x_3\quad\quad&=& 6 \nonumber \\
        -2x_1+3x_2+11x_3&=& 15 \nonumber \\
        -4x_1+3x_2+7x_3&=& 9 \nonumber
      \eea
      \ben
        \item
          % Question 1a
            Write the augmented matrix and transform it to reduced echelon form:
            \vspace{4in}
        \item
          % Question 1b
            Write the {\em general} solution to the linear system, if there is one:
      \een
    \newpage %%%%%%%%%%%%%%%%

  \item
True or false (no working needed, just circle the answer):
% will be one pt each

\vg

\ben
\itemsep 10ex

\item T / F: \qquad
\parbox{4in}{
The matrix $\left[\begin{array}{rr}1&1\\1&0\\0&1\end{array}\right]$
is in echelon form.
}
\item T / F: \qquad
\parbox{4in}{
  If an echelon form of an augmented matrix has a row of all zeros (including
  the right-hand side), then the linear system must be consistent.
}
\item T / F: \qquad
\parbox{4in}{
  Two linear systems that are row equivalent always reduce to the same
  reduced echelon form.
}
\item T / F: \qquad
\parbox{4in}{
Given any two vectors $\mbf{u}$ and $\mbf{v}$ in $\RR^m$, it always holds that
  Span$\{\mbf{v},\mbf{v},\mbf{u}\} = $ Span$\{\mbf{u},\mbf{v}\}$.
}
\item T / F: \qquad
\parbox{4in}{
A linear system with more equations than unknowns cannot have a unique solution.
}
\item T / F: \qquad
\parbox{4in}{
  The span of two vectors in $\RR^m$ is either a plane passing through the
  origin, a line passing through the origin, or the set consisting of the
  origin alone.
}
\een

\vspace{1in}

BONUS: Suppose two $1\times 2$ linear systems have the same {\em solution sets}.
Either prove that their augmented matrices have exactly the same reduced echelon form, or find a counterexample. [This is a challenge problem inspired by a student question; you may need more space!]
% This challenge problem helps advance us towards an answer to Andy's question in Lec 2.


      % Question 2
    \newpage %%%%%%%%%%%%%%%%

  \item
    % Question 3
    For what value(s) of the real number $h$ can the vector
    $\left[\begin{array}{r}1\\-1\\-5\end{array}\right]$
be written as a linear combination of the three vectors
$\left[\begin{array}{r}1\\1\\1\end{array}\right]$, 
$\left[\begin{array}{r}0\\1\\h\end{array}\right]$, and 
$\left[\begin{array}{r}2\\1\\-1\end{array}\right]$ ?

\vspace{5in}

For what value(s) of $h$ is the span of the set of three vectors equal
to $\RR^3$ ?
    
  \een
\end{document}
