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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 8
    (Monday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

Throughout, $X$ is assumed to be a vector space of dimension
$n<\infty$.

\begin{enumerate}


  % Lax Exercise 3.10, 3.11
\item Let $A,B\colon X\to X$ be linear maps.
  \begin{enumerate} 
  \item Show that if $A$ is invertible and similar to $B$, then $B$ is
    also invertible, and $B^{-1}$ is similar to $A^{-1}$.
  \item Show that if either $A$ or $B$ is invertible, then $AB$ and
    $BA$ are similar. \\
  \end{enumerate}



  % Lax Exercise 3.14
\item Suppose $T\colon X\to X$ is a linear map of rank 1.
  \begin{enumerate}
  \item Show that there exists $c\in K$ such that $T^2=cT$.
  \item Show that if $c\neq 1$, then $I-T$ has an inverse. \\
  \end{enumerate}



% Lax Exercise 3.15
  \item Suppose that $S,T\colon X\to X$ are linear maps.
  \begin{enumerate}
    \item Show that $\rank(S+T)\leq\rank(S)+\rank(T)$.
    \item Show that $\rank(ST)\leq\rank(S)$.
    \item Show that $\dim(N_{ST})\leq\dim N_S+\dim N_T$.
  \end{enumerate}
  For each of these, give an explicit example showing how equality
  need not hold. \\



\item Let $X$ and $U$ be vector spaces, and suppose that $Y$ is a
  subspace of $X$. Let $Q\colon X\to X/Y$ be the canonical quotient
  map sending $x\stackrel{Q}{\longmapsto}\{x\}$, and let $T\colon
  X\to U$ be a linear map. Give necessary and sufficient conditions
  for the existence of a unique linear map $S\colon X/Y\to U$ such
  that $T=S\circ Q$. When this happens, the map $T$ is said to \emph{factor
    through} the quotient space, as shown by the following commutative
  diagram:
  \[
  \xymatrix{X\ar[rr]^T\ar[dr]_Q && U \\ & X/Y\ar@{-->}[ur]_S}
  \]
  Prove all of your claims. \\


\end{enumerate}

\end{document}
