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\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 6, due Fri August 4}
        \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}
  \begin{enumerate}
    \item
      % Question 1
      Let $T:\RR^5\to\RR^3$ be a linear transformation with standard matrix $A$.
      \bigskip
      \begin{enumerate}
        \item
          What are the possible values for the rank of $A$?
          \vspace{1in}
        \item
          What are the possible values for the dimension of $\mathrm{Nul}\,A$?
          \vspace{1in}
        \item
          Suppose now that the $T$ from above is also onto.
          What are the possible values for the dimension of $\mathrm{Nul}\,A$?
          \vspace{1in}
        \item
          Let
          $
          A =
          \left[
            \begin{array}{rrrrr}
              2&0&-1&2&3\\
              4&0&-2&4&6\\
              0&0&1&-1&0
            \end{array}
          \right].
          $
          Find bases for $\mathrm{Col}\,A$, $\mathrm{Row}\,A$, and $\mathrm{Nul}\,A$ (In this question you do not need to show your working.)
          \vfill
      \end{enumerate}
      \pagebreak
    \item
      % Question 2
      Let
      $
        A=
        \left[
          \begin{array}{rrrr}
            -2&0&0&0\\
            0&-2&0&0\\
            0&0&3&1\\
            0&0&0&3
          \end{array}
        \right]
      $
       and 
      $
        B=
        \left[
          \begin{array}{rrrr}
            -2&0&0&0\\
            0&-2&0&0\\
            0&0&3&0\\
            0&0&0&3
          \end{array}
        \right]
      $.
      \bigskip
      \begin{enumerate}
        \item
          $A$ and $B$ have the same characteristic polynomial.
          Find this polynomial and explain what form and/or features
          that $A$ and $B$ have in common make the polynomial the same.
          \vspace{1in}
        \item
          From the previous part we see that each matrix has exactly 2 eigenvalues
          (call them $\lambda_1$ and $\lambda_2$ with $\lambda_1<\lambda_2$).
          Compute the dimensions of both eigenspaces for both of the matrices $A$ and $B$.
          Use this information to fill in the table below.
          Be sure to include $\lambda_1$ and $\lambda_2$ in the table.
          It is not necessary to include \emph{all} the computations involved,
          but please describe how you computed the dimensions.
          \bigskip
          \begin{center}
            \begin{tabular}{|c|c|c|}
              \hline
              matrix&dimension of $\lambda_1 = \;\;\;\;\;$ eigenspace&dimension of $\lambda_2 = \;\;\;\;\;$ eigenspace\\
              \hline
              $A$&&\\
              \hline
              $B$&&\\
              \hline
            \end{tabular}
          \end{center}
          \vspace{2in}
        \item Find a basis for the $\lambda_2$ eigenspace of $B$:
          \vfill
          % 2 pts
      \end{enumerate}
      \vfill
      \pagebreak
    \item
      % Question 3
      Let
      $
      A=
      \left[
        \begin{array}{rr}
          7&h\\
          2&11
        \end{array}
      \right]
      $.
      \bigskip
      \begin{enumerate}
        \item
          For what value(s) of $h$ does $A$ have an eigenvalue of (algebraic) multiplicity two?
          \vfill
        \item
          Let $\lambda$ be the eigenvalue of (algebraic) multiplicity two from the previous part.
          Compute a basis for the $\lambda$ eigenspace of $A$.
          \vfill
        \item
          Let
          $
          P=
          \left[
            \begin{array}{rr}
              -2&1\\
              2&0
            \end{array}
          \right]
          $,
          and let $B = P^{-1}AP$ (with $A$ defined by $h$ from part (a)).
          Compute $B$ and prove that $B$ and $A$ have the same eigenvalues.
          \vfill
      \end{enumerate}
  \end{enumerate}
\end{document}
