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\lhead{{\sf Summer Bridge Course (Algebra)}: Friday July 31}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesdsay, August 4 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Singular value decomposition, pseudo-inverses. \medskip

\noindent \textbf{Do}: Answer the following questions. Assume that all matrices are over the field \(K=\mathbb{C}\). 

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

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

%% HW 18, Problem 1
\begin{exercise}
 Consider the following matrices
 \[
  A=\begin{bmatrix}4&4\\-3&3\end{bmatrix},\qquad
  A^T\!A=\begin{bmatrix}25&7\\7&25\end{bmatrix},\qquad
  AA^T=\begin{bmatrix}32&0\\0&18\end{bmatrix}.
  \]
  \begin{enumerate}[label=(\roman*)]
  \item Find the eigenvalues \(\sigma_1^2\), \(\sigma_2^2\) and unit
    eigenvectors \(v_1\), \(v_2\) of \(A^TA\). 
  \item For the \(\sigma_i\neq 0\), compute \(u_i=Av_i/\sigma_i\) and
    verify that indeed \(||u_i||=1\). Find the other \(u_i\) by computing
    the other unit eigenvector of \(AA^T\).
  \item Construct the singular value decomposition (SVD),
    \(A=U\Sigma V^T\).
  \item Write down orthonormal bases for each the ``four fundamental
    subspaces'': the column space \(R_A\), the nullspace \(N_A\), the row
    space \(R_{A^T}\), and the left nullspace \(N_{A^T}\).
  \item Describe \emph{all} matrices that have the same four fundamental
    subspaces.
  \item Find a left, right, and pseudoinverse of \(A\), or explain why
    it doesn't exist. 
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

  %%------------------------------------------------------------------------
  
  %% HW 18, Problem 2
  \begin{exercise}
    Consider the matrix
    \(
    A=\begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}.
    \)
    \begin{enumerate}[label=(\roman*)]
    \item Construct the singular value decomposition of \(A\).
    \item Write down orthonormal bases for each the ``four fundamental
      subspaces'': the row space \(R_A\), the nullspace \(N_A\), the column
      space \(R_{A^T}\), and the left nullspace \(N_{A^T}\).
    \item Find a left inverse, right inverse, and pseudoinverse of \(A\),
      or explain why it doesn't exist. 
    \end{enumerate}
  \end{exercise}
  
%%------------------------------------------------------------------------

%% HW 18, Problem 3
\begin{exercise}
  Let \(f\colon\R^4\to\R^3\) be the linear map defined by \(f(x) =
  Mx\) for \(x\in\R^4\) where \(M=ABC\) and
  \[
  A = \begin{bmatrix} 1  & 1 & 1 \cr 1 & 1 & -1 \cr 1 &-2&0  \cr  \end{bmatrix}, \quad 
  B =  \begin{bmatrix}\frac{-1}{\sqrt{3}} & 0 & 0 & 0  \cr 0 & \frac{1}{\sqrt{6}} & 0 &0 \cr  0 &0  & -\sqrt{2} & 0 \cr    \end{bmatrix}, \quad 
  C =  \begin{bmatrix}1 & 1 & 1 & 1 \cr 1 & 1 & -1 & -1 \cr  1 & -1&1  & -1 \cr   1 & -1& -1 & 1 \cr  \end{bmatrix}.
  \]
  \begin{enumerate}
  \item Define the adjoint map \(f^*\colon\R^3 \to \R^4\) (under the standard Euclidean inner product) and express it in terms of \(M\).
  \item Find a singular value decomposition (SVD) of \(M\).
    (\emph{Hint}: Observe that \(A^TA\)  and \(C^TC\) are diagonal.)
  \item Find all \(x\in\R^4\) with \(\|x\|=1\) so that \(\|Mx\|\) is maximized.  
  \item Describe the eigenvalues and  eigenvectors of \(M^TM\).
  \item Find the least square solution for \(Mx=b\)  with  \(\|x\|_2\) minimal where \(b=(1,1,1)^T\). (\emph{Hint}: Use the pseudo-inverse of \(M\).)
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

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

\end{document}
