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\begin{document}

\begin{center}
{\Large TEST 2

Math 315}
\end{center}

\begin{itemize}
\item  Show all your work and make your work neat.  You will be graded on this!
\item  You will be graded on both your answer and the work done to get that answer.
\item  You can use your calculator, but I should be able to follow your work as if you had no calculator!
\item  Do your work on other paper.
\item  There are 5 questions worth 10 points each and 1 extra credit problem.
\end{itemize}


\begin{enumerate}

\item  Consider the matrix $A$ below.

  \[
  A = \left[
    \begin{array}{rrrrr}
      1 & 0 & 3 & 4 & 0 \\
      0 & 1 & 8 & -1 & 1 \\
      1 & 1 & 11 & 3 & 1
    \end{array}
    \right]
  \]
  \begin{enumerate}
  \item  What is the rank of $A$? (justify your answer)
  \item  \label{row space}
    Find a basis for the row space of $A$. (justify your answer)
  \item  Find a basis for the column space of $A$. (justify your answer)
  \item  What is the dimension of the null space of $A$? (justify your answer)
  \item  Given that the vector $\mathbf{v}=[8, -5, -16, 37, -5]^t$ is in the row space of $A$, 
    write $\mathbf{v}$ in the coordinates determined by your basis in part~\ref{row space}.
  \end{enumerate}

\item  Consider the subspace of $\R^4$:
  \[
  W = \mathrm{Span}\{ [4,-5,1,1]^t \}
  \]
  Let $v=[-12,8,4,-2]^t$.
  \begin{enumerate}
  \item  Find the orthogonal projection of $\mathbf{v}$ onto the subspace $W$, 
    $\mathrm{Proj}_W(\mathbf{v})$.
  \item  Determine a vector, $\mathbf{u}$, orthogonal to $W$ so that 
    $\mathrm{Proj}_W(\mathbf{v}) + \mathbf{u} = \mathbf{v}$.
    Show that the vector you found is orthogonal to $W$.
  \end{enumerate}

\item  Let $A$ be an arbitrary $3 \times 2$ matrix.
  \begin{enumerate}
  \item  $A$ determines a linear transformation from $\R^n$ to $\R^m$, what is $n$ and $m$?
  \item  Determine all possibilities for the Rank of $A$ and nullity of $A$.  (justify your answer)
  \item  Determine all possibilities for the Rant of $A^t$ and nullity of $A^t$, the transpose of $A$.
    (justify your answer).
  \end{enumerate}

\newpage

\item  \label{p trans}
  Suppose $T:\R^2 \ra \R^3$ is a linear transformation such that
  \[
  T\left( \left[
    \begin{array}{r}
      1 \\ 0
    \end{array}
    \right]
  \right)
  = 
  \left[
    \begin{array}{r}
      1 \\ 2 \\ -5
    \end{array}
    \right]
  \qquad
  T\left( \left[
    \begin{array}{r}
      2 \\ 1
    \end{array}
    \right]
  \right)
  = 
  \left[
    \begin{array}{r}
      -1 \\ 0 \\ 1
    \end{array}
    \right]
  \]
  \begin{enumerate}
  \item  Write the vector $[0,1]^t$ as a linear combination of $[1,0]^t$ and $[2,1]^t$.
    \\
    (This isn't supposed to be very hard, you should be able to do this without solving equations.
    But, I certainly won't stop you from solving equations.)
  \item  Using the combination found above, find $T([0,1]^t)$
  \item  \label{trans}
    Using this information, find a matrix that represents the transformation $T$.
  \end{enumerate}

\item
  Consider the standard basis for $\R^2$ and the basis $B=\{ [1,0]^t, [2,1]^t \}$.
  \begin{enumerate}
  \item  \label{p matrix}
    Find the change of coordinate matrix to transform coordinate in $B$ to standard coordinates.
    Call this matrix $P$.
    (the point matrix using the text's language).
  \item  
    Find the change of coordinate matrix to transform standard coordinates to coordinates in $B$.
    Call this matrix $C$.
    (the coordinate matrix using the text's language).
  \end{enumerate}

\item  (Extra credit - 5 points)
  Let $A$ be the matrix from part~\ref{trans} and $P$ the matrix from part~\ref{p matrix}.
  \begin{enumerate}
  \item  Determine the product $AP$.  
  \item  $AP$ should be a matrix with columns equal to the vectors in problem~\ref{p trans}.
    Explain why this happens.
  \end{enumerate}


\end{enumerate}

\end{document}