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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 5, due Fri July 28}
        \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
      $
      A=
      \left[
        \begin{array}{rrrr}
          5&-1&-3&11\\ -2&1&0&-5\\3&-2&1&8
        \end{array}
        \right].
      $
      \begin{enumerate}
        \item
          Find a basis for $\mathrm{Col}\,A$.
          \vspace{2.5in}
        \item
          What is the dimension of the subspace spanned by the columns of $A$ ?
          \vspace{.3in}
        \item
          Find a basis for $\mathrm{Nul}\,A$.
          \vspace{2.5in}
        \item
          Write, and explain, an
          inequality satisfied by the dimension of the nullspace
          of {\em any} matrix $B$ formed by deleting rows from $A$. [Hint:
            what is a subspace of what?]
      \end{enumerate}
      \pagebreak
    \item
      % Question 2
      Consider a linear transformation $T:V\to W$
      where $V$ is a vector space with basis $\mathcal{B} = \{\mbf{b}_1,\dots,\mbf{b}_n\}$
      and $W$ is a vector space with basis $\mathcal{C} = \{\mbf{c}_1,\dots,\mbf{c}_m\}$.
      \bigskip
      \begin{enumerate}
        \item
          Let $\mbf{x},\mbf{y}$ be in $V$.
          Prove that $[\mbf{x}+\mbf{y}]_\mathcal{B} = [\mbf{x}]_\mathcal{B}+[\mbf{y}]_\mathcal{B}$.  (Here you may use results up to and including the Unique Representation Theorem. Keep in mind $V$ can be any vector space, not necessarily $\mathbb{R}^n$.)
          \vspace{3in}
        \item
          Suppose the map $T$ is onto.
          What relation between $n$ and $m$ must hold?
          Prove your answer. [Hint: construct a map from
            $\mathbb{R}^n$ to $\mathbb{R}^m$ using coordinate maps, then prove
          that map is onto]
          \vfill
      \end{enumerate}
      \pagebreak
    \item
      % Question 3
      \ben
    \item Recall the standard basis $\{1,t,t^2\}$ for $\mathbb{P}_2$.
      When we express these monomials about a new origin $t=a$,
      we get the set $\{1,(t-a), (t-a)^2\}$.
      Prove, for any $a$, that this set is also a basis for $\mathbb{P}_2$.
      [Hint: you many use that the coordinate map for the standard basis
        is an isomorphism.]
      \vspace{3.5in}
      
      \item 
        The subspace $H = $ Span
        $\left\{\left[\begin{array}{r}1\\1\end{array}\right]\right\}$ is
        in the span of the vectors
        $\mbf{b}_1 = \left[\begin{array}{r}1\\0\end{array}\right]$
        and $\mbf{b}_2 = \left[\begin{array}{r}0\\1\end{array}\right]$.
        Explain whether $\{\mbf{b}_1,\mbf{b}_2\}$ is a basis for $H$.
        
      \een
  \end{enumerate}
\end{document}
