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% 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 2, due Fri July 7}
        \quad
    {\em Please show your work, and check your answers. No credit is given for solutions without work or justification.}
    \vskip .25in
  % NUMBERED QUESTIONS
  \thispagestyle{empty} % do we want this?
  \begin{enumerate}
    \item
      % Question 1
      Let $h$ be a scalar, and consider the set of three vectors:
      \begin{equation*}
        \mbf{v}_1
        =
        \left[
          \begin{array}{r}
            3\\-2\\-4
          \end{array}
        \right]
        ,\;
        \mbf{v}_2
        =
        \left[
          \begin{array}{r}
            -6\\3\\9
          \end{array}
        \right]
        ,\;
        \mbf{v}_3
        =
        \left[
          \begin{array}{r}
            -3\\2(h-1)\\7
          \end{array}
        \right].
      \end{equation*}
      \bigskip
      \begin{enumerate}
        \item
          % Question 1a
          With $h=1/2$,
          is the set linearly independent,
          and why?
          If not,
          give a dependence relation
          between $\mbf{v}_1$, $\mbf{v}_2$, and $\mbf{v}_3$.
          \vfill
        \item
          % Question 1b
          With $h=0$,
          is the set linearly independent,
          and why?
          If not,
          give a dependence relation
          between $\mbf{v}_1$, $\mbf{v}_2$, and $\mbf{v}_3$.
          \vfill
      \end{enumerate}
    \newpage %%%%%%%%%%%%%%%%
    \item
      % Question 2
      Consider the following network flow diagram
      where the numbers indicate known flows,
      and $x_1,x_2, x_3,x_4$ indicate unknown flows
      on their respective edges.
      \begin{center}
        \begin{tikzpicture}[node distance = 1.8 cm]
          %\useasboundingbox (-1,-1) rectangle (11,11);
          \tikzset{VertexStyle/.style = {shape = circle, fill = blue!20, text = black, inner sep = 2pt, outer sep = 0pt, minimum size = 16pt}}
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          \tikzset{LabelStyle/.style = {fill = white, text = black}}
          %\tikzset{LabelStyle/.style = {draw, fill = white, text = black}}
          \node[VertexStyle](A){A};
          \node[left=of A](Aout){};
          \node[below left=of A](Ain){};
          \draw[EdgeStyle](Ain) to node[LabelStyle]{$x_4$} (A);
          \draw[EdgeStyle](A) to node[LabelStyle]{$1$} (Aout);
          \node[VertexStyle,above right=of A](B){B};
          \node[above left=of B](Bin){};
          \node[above right=of B](Bout){};
          \draw[EdgeStyle](Bin) to node[LabelStyle]{$2$} (B);
          \draw[EdgeStyle](B) to node[LabelStyle]{$3$} (Bout);
          \draw[EdgeStyle](A) to node[LabelStyle]{$x_2$} (B);
          \node[VertexStyle,below right=of B](C){C};
          \node[right=of C](Cin){};
          \node[below right=of C](Cout){};
          \draw[EdgeStyle](C) to node[LabelStyle]{$5$} (Cout);
          \draw[EdgeStyle](Cin) to node[LabelStyle]{$4$} (C);
          \draw[EdgeStyle](B) to node[LabelStyle]{$x_3$} (C);
          \draw[EdgeStyle](C) to node[LabelStyle]{$x_1$} (A);
          %\tikzset{EdgeStyle/.append style = {bend left}}
        \end{tikzpicture}
      \end{center}
      \begin{enumerate}
        \item
          % Question 2a
          Find the \emph{reduced} echelon form for the
          corresponding linear system.
          \vfill
        \item
          % Question 2b
          If a solution exists,
          write the \emph{parametric vector form}
          of the solution set.
          \vfill
        \item
          % Question 2c
          Assume all flows are nonnegative.
          What constraint does this impose on $x_3$?
      \end{enumerate}
    \newpage %%%%%%%%%%%%%%%%
    \item
      % Question 3
      \begin{enumerate}
        \item
          % Question 3a
          Let $T:\RR^3\to\RR^3$ be a linear transformation defined by
          \begin{equation*}
            T(\mbf{x}) = x_1\mbf{v}_1+x_2\mbf{v}_2+x_3\mbf{v}_3,\quad\text{ with}
          \end{equation*}
          \begin{equation*}
            \mbf{x}
            =
            \left[
              \begin{array}{r}
                x_1\\x_2\\x_3
              \end{array}
            \right]
            ,\quad
            \mbf{v}_1
            =
            \left[
              \begin{array}{r}
                3\\-2\\5
              \end{array}
            \right]
            ,\quad
            \mbf{v}_2
            =
            \left[
              \begin{array}{r}
                4\\-3\\1
              \end{array}
            \right]
            ,\text{ and }\quad
            \mbf{v}_3
            =
            \left[
              \begin{array}{r}
                7\\1\\2
              \end{array}
            \right].
          \end{equation*}
          Find a matrix $A$ such that
          $T(\mbf{x}) = A\mbf{x}$
          for every $\mbf{x}$ in $\RR^3$.
          Explain why your answer is correct.
          \vfill
        \item
          % Question 3b
          Let $\mbf{p}$ be any vector in $\RR^n$
          and let $\mbf{v}$ be any nonzero vector in $\RR^n$.
          Consider the line $\ell$ defined by the parametric equation
          \begin{equation*}
            \mbf{x} = \mbf{p}+t\mbf{v},\quad t\text{ in }\RR.
          \end{equation*}
          Let $T:\RR^n\to\RR^n$ be a linear transformation.
          Show that the image of $\ell$ under the map $T$
          is either a line in $\RR^n$ or a single point.
          \vfill
      \end{enumerate}
  \end{enumerate}
\end{document}
