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\begin{document}
\centerline{Math 201 Homework for Friday, Week 7}
\bigskip
\textbf{Due:} Friday, October 15.

\begin{prob}
   The \emph{trace} of an~$n\times n$ matrix~$A$ is the sum of its diagonal
      elements:
      \[
        \tr(A)=\sum_{i=1}^nA_{ii}.
      \]
      \begin{enumerate}
        \item If~$A$ and~$B$ are~$n\times n$ matrices, prove
  	that~$\tr(AB)=\tr(BA)$. (Use the definition of matrix multiplication and
  	  summation notation in your proof.)
  
  
        \item If~$P$ is an invertible~$n\times n$ matrix, prove
  	that~$\tr(PAP^{-1})=\tr(A)$.
  	      \item Consider the following ordered basis
  	for~$\mathcal{M}_{2\times 2}(F)$:
  	\[
  \alpha=
  \left\langle
  \left(\begin{array}{cc}
      1&0\\0&0
  \end{array} \right),\quad
  \left(\begin{array}{cc}
      0&1\\0&0
  \end{array} \right),\quad
  \left(\begin{array}{cc}
      0&0\\1&0
  \end{array} \right),\quad
  \left(\begin{array}{cc}
      0&0\\0&1
  \end{array} \right)
  \right\rangle.
  	\]
      \end{enumerate}
  The trace defines a function~$\tr\colon\mathcal{M}_{2\times 2}(F)\to F$, and
  it is not hard to check that it is a linear transformation of vector spaces
  over $F$ (you don't have to prove that here, although it is a good exercise to
  do).  Compute the matrix representing the trace
  function~$\tr\colon\mathcal{M}_{2\times 2}(F)\to F$ with respect to~$\alpha$
  for the domain and with respect to the basis~$\left\{ 1 \right\}$ for the
  codomain.
\end{prob}

\begin{prob}
  Let 
  \[A=\begin{pmatrix} 1 & 3\\ 2& -1\end{pmatrix}, \quad B=\begin{pmatrix} 1 & 0 & -3\\ 4 & 1 & 2\end{pmatrix}, \quad C=\begin{pmatrix} 1 & 1 & 4\\ -1 & -2 & 0\end{pmatrix}, \quad D=\begin{pmatrix} 2 \\ -2 \\ 3\end{pmatrix}.\]
  Compute, if possible, the following. If it is not possible, explain why.
  \begin{enumerate}
   \item $AB$,
      \item $A(2B+3C)$,
     \item $(AB)D$,
   \item $A(BD)$,
   \item $AD$.
     \end{enumerate}
\end{prob}

\begin{prob}
  Let $V$ be a vector space over a field $F$. Recall that the identity function
  $\id_V \colon V \to V$ is given by $\id_V(v)=v$ for all $v\in V$. This
  function is linear (if you are not convinced, prove it, but you do not have to
  turn in that proof).
  \begin{enumerate}
   \item Let $V$ be a vector space of dimension $n$ and let $\mathcal{B}$ be an
     ordered basis for $V$. Show that the matrix representing $\id_V$ with
     respect to the basis $\mathcal{B}$ for both the domain and the codomain is
     $I_n$ (the $n\times n$ identity matrix).
   \item Let $V$ and $W$ be vector spaces of dimension $n$ and let $f\colon V
     \to W$ be an isomorphism with inverse $f^{-1} \colon W \to V$. Let
     $\mathcal{B}$ and $\mathcal{D}$ be ordered bases for $V$ and $W$,
     respectively. If $A$ is the matrix representing $f$ with respect to the
     bases $\mathcal{B}$ and $\mathcal{D}$, what is the matrix for $f^{-1}$ with
     respect to the bases $\mathcal{D}$ and $\mathcal{B}$? Justify your answer.
      \item Consider the linear transformation $f\colon \R^2 \to \R^2$ given by
	$f(x,y)=(3x+y,-x+4y)$. Using part (b), find the inverse of $f$.
  \end{enumerate}
\end{prob}
\end{document}

