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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 3, due Fri July 14}
        \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 $T:\RR^2\to\RR^3$ be a linear transformation defined by
      \[
        T
        \left(
          \left[
            \begin{array}{r}
              x_1\\
              x_2
            \end{array}
          \right]
        \right)
        =
        \left[
          \begin{array}{r}
            x_1-2x_2\\
            -x_1+3x_2\\
            3x_1-2x_2
          \end{array}
        \right].
      \]
      \bigskip
      \begin{enumerate}
        \item
          % Question 1a
          Find the standard matrix for $T$.
          \vfill
        \item
          % Question 1b
          Is $T$ one-to-one? Explain why or why not.
          \vfill
        \item
          % Question 1c
          Is $T$ onto? Explain why or why not.
          \vfill
      \end{enumerate}
      \newpage
    \item
      % Question 2
      Let
      $
      A =
      \left[
        \begin{array}{rrr}
          \mbf{a}_1&\mbf{a}_2&\mbf{a}_3
        \end{array}
      \right]
      =
      \left[
        \begin{array}{rrr}
          -2&1&-4\\
          1&0&2\\
          0&-2&1
        \end{array}
      \right].
      $
      \bigskip
      \begin{enumerate}
        \item
          % Question 2a
          If $A$ is not invertible, prove it, else, if $A$ is invertible, find $A^{-1}$.
          \vfill
        \item
          % Question 2b
          Use the previous part of this problem to express the vector
          $
          \left[
            \begin{array}{r}
              -2\\
              3\\
              1
            \end{array}
          \right]
          $
          as a linear combination of the columns of $A$.
          \vfill
        \item
          % Question 2c
          Let $A$ be an arbitrary $4\times 4$ invertible matrix.
          Write a linear system involving the matrix $A$ whose solution gives
          the third column of $A^{-1}$.
          (Write your linear system as a matrix equation).
          \vfill
      \end{enumerate}
      \newpage
    \item
      % Question 3
      A certain disease affects the human population as follows over the
      course of one year:
      Two thirds of the initially healthy (H) humans become sick (S), while
      the rest remain healthy.
      Over that same year, one third of the initially sick recover
      and become healthy, one third remains sick, and one third sadly dies.
      \bigskip
      \begin{enumerate}
        \item
          % Question 3a
          Stacking the H and S populations into a column vector $\mbf{x}$,
          write the population vector at year $k+1$ in terms of that at year $k$.
          Be sure to give the matrix. [Note: the dead are not included in the column
          vector nor the matrix.]
          \vfill
        \item
          % Question 3b
          If the populations at ``year zero'' are 900 healthy and zero sick,
          what are the populations at year 2? (ie, after two years of dynamics).
          \vfill
        \item
          % Question 3c
          Find a {\em matrix} that takes any population vector at
          year $k$ to that at year $k+2$. [Hint: check against part b.]
          \vfill
      \end{enumerate}
  \end{enumerate}
\end{document}
