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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due November 13 
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2020 $|$ M.~Macauley}}


\begin{document}

$\;$

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

\begin{enumerate}

%%------------------------------------------------------------------------ 
  
\item Use the Gram-Schmidt process to find an orthonormal basis for
  the subspace of $\R^4$ spanned by $y_1=(1,2,1,1)$, $y_2=(1,-1,0,2)$
  and $y_3=(2,0,1,1)$. \\

%%------------------------------------------------------------------------ 

\item Let $X$ be the vector space of all continuous real-valued
  functions on $[0,1]$. Define an inner product on $X$ by
  \[
  (f,g)=\int_0^1 f(t)g(t)\,dt\,.
  \]
  Let $Y$ be the subspace of $X$ spanned by $f_0,f_1,f_2,f_3$, where
  $f_k(x)=x^k$. Find an orthonormal basis for $Y$. \\

%%------------------------------------------------------------------------ 

\item Let $Y$ be a subspace of an inner product space $X$, and $P_Y\colon
  X\to X$ the orthogonal projection onto $Y$. Prove that $P_Y^*=P_Y$. \\

%%------------------------------------------------------------------------ 

\item Let $X$ be a finite-dimensional real inner product 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 finite-dimensional
  inner product 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}

%%------------------------------------------------------------------------ 

\end{enumerate}

\end{document}







  
