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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 4, due Fri July 21}
        \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
      \ben
    \item Find the determinant of the matrix
      $A = \left[\begin{array}{cccc} 1&0&2&3 \\ 2&0&4&7\\
        1&0&5&4 \\ 17&2&38&99
      \end{array}\right]$
      \vspace{3.5in}
      \item Is $A$ from above invertible, and why?

        \vspace{1in}
        
      \item Prove that, if $A$ is any $n\times n$ matrix and $P$ is an
        invertible $n\times n$ matrix, that \\ det $(PA P^{-1}) = $ det $A$.
        [Please state when and which properties of det you use.]

          \een

          \newpage

  \item % Question 2
Is each of the following sets a vector space? Explain what result(s) you
used to prove your claim. You may assume that $\RR^n$ is a vector space.
\ben
\item
  $\left\{ \left[\begin{array}{c}a\\b\\c\end{array}\right], \quad
  a + b + c = 0  \quad \mbox{and} \quad 3a -b + 2c = 0  \right\}$

\vspace{1.8in}

\item The set of vectors
  $\left[\begin{array}{c}2\\1\end{array}\right] +
  t \left[\begin{array}{c}1\\3\end{array}\right]$,
  where $t$ takes all real values.

\vspace{1.8in}

\item The set of vectors
  $\left[\begin{array}{c}2\\1\end{array}\right] +
  t \left[\begin{array}{c}6\\3\end{array}\right]$,
  where $t$ takes all real values.

\vspace{1.8in}

\item The set of functions of the form $a + bt^2$, where $a$ and $b$
  are real. [Hint: relate this set to $\mathbb{P}_2$]
\een

    \pagebreak
          
    \item   % Question 3
      Let $T:\RR^4\to\RR^3$ be a linear transformation with standard matrix
      \[
        A =
        \left[
          \begin{array}{rrrr}
            3&0&6&-3\\
            0&1&-1&0\\
            1&0&2&-1
          \end{array}
        \right].
      \]
      \bigskip
      \begin{enumerate}
        \item
          Write the kernel of $T$ (null space of $A$) in form of a span.
          \vspace{2.5in}

        \item Is Col $A$ equal to $\RR^3$ ? (explain)
          
            \vspace{2in}

          \item True/False: If the kernel of any linear transformation
            $T:\RR^n\to\RR^m$ is a single point, then $n \ge m$ ? (explain, or, if false, correct the deduction)
           

\vspace{2in}
            
          \item[BONUS] Find a $2\times 2$ matrix $A$ for whom
            Col $A$ and Nul $A$ are equal (careful: equal as sets).
            
      \end{enumerate}
  \end{enumerate}
\end{document}
