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\begin{document}
\centerline{Math 201 Homework for Tuesday, Week 5}
\bigskip
{\bf Due:} Tuesday, September 28.

\begin{prob}
  Let~$A$ be an~$m\times n$ matrix with~$i,j$-th entry~$A_{ij}$.
  The {\em tranpose} of~$A$, denoted~$A^{T}$, is the~$n\times m$ matrix
  with~$i,j$-th entry~$A_{ji}$: the~$i$-th row of~$A^{T}$ is the~$i$-th column
  of~$A$.  Thus, for example,
  \[
    \left(\begin{array}{cc}
	a&b\\
	c&d\\
	e&f
    \end{array} \right)^{T}
    =
    \left(\begin{array}{ccc}
	a&c&e\\
	b&d&f
    \end{array} \right).
  \]
  A matrix~$A$ is {\em skew symmetric} if~$A^{T}=-A$ (notice the minus sign!).   

  Let~$W$ be the set of~$3\times 3$ skew symmetric matrices over $\R$.
  \begin{enumerate}
    \item Prove that~$W$ is a subspace of the vector space of all~$3\times 3$ matrices
      over~$\R$.
    \item Give a basis for~$W$.
    \item What is~$\dim(W)$?
  \end{enumerate}
\end{prob}

\begin{prob}
  Define the following matrix over the real numbers:
  \[
    M = 
    \left(\begin{array}{rrrr}
	-14 & 56 & 40 & 92 \\
	6 & -24 & -17 & -39 \\
	8 & -32 & -23 & -53 \\
	-1 & 4 & 3 & 7
    \end{array}\right).
  \]
  \begin{enumerate}
    \item What is the reduced echelon form for~$M$? (You do not need to show
      your work for this.)
    \item Compute (i) a basis for the row space of~$M$ and (ii) a basis for
      the column space of~$M$ \emph{using the algorithm presented in class} on
      Friday of Week~$4$. (There is a unique solution if you use the algorithm.)
  \end{enumerate}
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item Prove that there exists a linear transformation $f\colon \R^2 \to
      \R^3$ such that $f(2,1)=(0,-1,3)$ and $f(-1,2)=(1,0,-4)$. What is
      $f(5,10)$?
    \item Is there a linear transformation $f\colon \R^3 \to \R^2$ such that
      $f(1,2,1)=(2,3)$, $f(3,1,4)=(6,2)$ and $f(7,-1,10)=(10,1)$? Explain your
      reasoning.
  \end{enumerate}
\end{prob}
\end{document}

