\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 3}

\begin{document}
  \maketitle

  \begin{defn*}
      The \emph{transpose} $A^\top$ of an $m\times n$ matrix $A$ is the $n\times m$ matrix with entries $(A^\top)_{ij} = A_{ji}$.  A square matrix $P$ is called a \emph{projection} if $P^2 = P$ and $P^\top = P$.
  \end{defn*}
  \begin{enumerate}
    \item  Show that $(AB)^\top = B^\top A^\top$ for all matrices $A$ and $B$ for which $AB$ is defined.  [10]
    \item  Prove that a square matrix $A$ is invertible if and only if $A^\top$ is invertible.  [10]
    \item  Prove the following theorem from class. [15]
    \begin{thm*}
      Let $a_1,\ldots,a_n\in\RR^m$ and let $A$ be the $m\times n$ matrix whose columns are $a_1,\ldots,a_n$.  Then the following are equivalent;
      \begin{enumerate}
        \item  $\lspan\{a_1,\ldots,a_n\} = \RR^m$,
        \item  $A$ is right-invertible.
      \end{enumerate}
    \end{thm*}
    \item  Let $P$ be an $n\times n$ diagonal matrix whose diagonal values are all either zero or one.  Prove that $P$ is a projection, and find bases for $\ran(P)$ and $\nul(P)$.  [10]
    \item  Show that there are $2\times 2$ projections $P$ and $Q$ such that $PQ \neq QP$, and $\tr(P) = \tr(Q)$.  [10]
    \item  Find bases for $\nul(A)$ and $\ran(A)$, where $A$ is the following matrix; [10]
    \[
      \begin{pmatrix}
        1 & 0 & 3 & -1 \\
        2 & 1 & 0 & 1 \\
        0 & 4 & 5 & 2 \\
      \end{pmatrix}
    \]
    \item  Find a basis for the subspace of $\RR^3$ spanned by the following set of vectors;  [10]
    \[
      \left\{
        \begin{pmatrix}
          0 \\ 1 \\ 1 \\ 2 \\
        \end{pmatrix},
        \begin{pmatrix}
          1 \\ 1 \\ 2 \\ 3 \\
        \end{pmatrix},
        \begin{pmatrix}
          1 \\ 2 \\ 3 \\ 5 \\
        \end{pmatrix},
        \begin{pmatrix}
          2 \\ 3 \\ 5 \\ 8 \\
        \end{pmatrix}
      \right\}
    \]
    \item  Let $A$ be an $n\times n$ matrix.  Show that the sequence of integers
    \[
      d_t = \dim\nul(A^t)
    \]
    is nondecreasing and eventually constant.  Show that if $d$ is the eventual value of this sequence, then $d_k = d$ for all $k\ge n$.  (That is, the sequence reaches this eventual value at $t = n$ at the latest.)  [20]

 \end{enumerate}

\end{document}
