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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due October 31
    (Monday) }} \rhead{\thepage} \cfoot{{\sf Math 853 $|$ Linear
    Algebra $|$ Fall 2016 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent\emph{Read}: Lax, Chapter 7, pages 89--100. 

\begin{enumerate}


% Lax Exercise 7.2 (p. 87)
\item Let $Y$ be a subspace of a Euclidean space $X$, and $P_Y\colon
  X\to X$ the orthogonal projection onto $Y$. Prove that $P_Y^*=P_Y$.


% Lax Exercise 7.7 
\item Let $X$ be a finite-dimensional real Euclidean space. We say
  that a sequence $\{A_n\}$ of linear maps converges to a limit $A$ if
  $\displaystyle\lim_{n\to\infty}||A_n-A||=0$.
  \begin{enumerate}
  \item Show that $\{A_n\}$ converges to $A$ if and only if for all
    $x\in X$, $A_nx$ converges to $Ax$.
  \item Show by example that this fails if $\dim X=\infty$.
  \end{enumerate}
  

\item Let $A\colon X\to U$ be a linear map between Euclidean spaces,
  and let $A^*\colon U\to X$ denote the adjoint map. The map $A$ has a
  \emph{left inverse} if there is a linear map $L\colon U\to X$ such
  that $LA=I_X$, the identity on $X$. It has a \emph{right inverse} if
  there is a linear map $R\colon U\to X$ such that $AR=I_U$ is the
  identity on $U$.
  \begin{enumerate}
  \item Prove that $R_{A^*}^\perp=N_A$.
  \item Prove that $A$ maps $R_{A^*}$ bijectively onto $R_A$. 
  \item Show that if $A$ has a left inverse, then $Ax=u$ has \emph{at
    most} one solution. Give a condition on $u$ that completely
    characterizes when there is a solution.
  \item Show that if $A$ has a right inverse, then $Ax=u$ has \emph{at
    least} one solution. If $Ax_p=u$ for some particular $x_p\in X$,
    then describe all solutions for $x$ in this case. What condition
    ensures that there will be \emph{only} one solution?
  \item What are the possibilities for the rank of $A$ if it has
    a left inverse? What if it has a right inverse?
  \end{enumerate}



  % Lax Exercise 7.15 (p. 96)
\item Let $X$ be the space of continuous complex-valued functions on
  $[-1,1]$ and define an inner product on $X$ by
  \[
  (f,g)=\int_{-1}^1f(s)\overline{g(s)}\,ds\,.
  \]
  Let $m(s)$ be a continuous function of absolute value $1$, that is,
  $|m(s)|=1$, $-1\leq s\leq 1$. Define $M$ to be multiplication by $m$:
  \[
  (Mf)(s)=m(s)f(s)\,.
  \]
  Show that $M$ is unitary.

  
  
  % July 2009 B1(a), and July 2008 B2(c)
\item Let $A$ be a linear map of a finite-dimensional complex
  Euclidean space $X$.
  \begin{enumerate}
  \item A matrix is \emph{normal} if $AA^*=A^*A$. It is unitarily
    similar to a diagonal matrix if $A=U^*DU$ for a diagonal matrix
    $D$ and unitary matrix $U$. Show that these conditions are
    equivalent.
  \item Prove that if $A$ is normal then it has a square-root, that
    is, a matrix $B$ such that $A=B^2$. Is $B$ necessarily normal?
    Unique?
  \item Suppose that $A$ is diagonalizable. Prove that $A$ is normal
    if and only if each eigenvector of $A$ is an eigenvector of $A^*$.
  \end{enumerate}
  


\end{enumerate}

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