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\lhead{{\sf Bridge course (Algebra)}: Monday July 29}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 30 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

\noindent \textbf{Topic}: Self-adjoint matrices, quadratic forms, left
\& right inverses. \\

\noindent \textbf{Read}: \emph{Spectral theorems and singular value decomposition (SVD)}, by Shuhong Gao. \\

\noindent \textbf{Do}: Answer the following question. \\

\begin{enumerate}

%%------------------------------------------------------------------------
 
\item Consider the quadratic form $q(x)=2x_1^2+6x_1x_2+2x_2^2$. 
  \begin{enumerate}
  \item Write this as $q(x)=x^TAx$, for some $A$.
  \item Write $A=PDP^T$, where $D$ is a diagonal matrix and $P$
    is orthogonal with determinant $1$, i.e., a rotation matrix.
  \item Change variables by letting $z=P^Tx$. Sketch the level curve
    $q(x)=1$ in both the $z_1z_2$-plane and in the $x_1x_2$-plane. \bigskip
  \end{enumerate}

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%
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%
%\begin{figure}[!ht]
%\includegraphics[width=.75\textwidth]{image.png}
%\caption{The graph of $q(x)=1$ in the $x_1x_2$-plane, and in the $z_1z_2$-plane.}\label{fig:quadratic-form}
%\end{figure}
  
 %%--------------------------------------------------------------------- 
  
\item Let $A\colon X\to U$ be a linear map between finite-dimensional
  vector spaces, with $\dim X=n$ and $\dim U=m$. 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 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? \bigskip
  \end{enumerate}

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

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

\end{enumerate}

\end{document}
