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\lhead{{\sf Summer Bridge Course (Algebra)}: Friday July 3}
\chead{}\rhead{\thepage}
\lfoot{Due Monday, July 6 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

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


\noindent \textbf{Topics}: The four fundamental subspaces. \\

\noindent \textbf{Read}: Pages 1--2 of \emph{The Four Fundamental Subspaces: 4 Lines} by Gil Strang. 
\begin{center}
\url{https://web.mit.edu/18.06/www/Essays/newpaper_ver3.pdf}
\end{center}

\noindent \textbf{Do}: Unless otherwise specified, let \(\A\) be an \(m\times n\) matrix, which we can associate with a linear map \(\R^n\to\R^m\). Let \(C(\A)\), \(C(\A^T)\), \(N(\A)\), and \(N(\A^T)\) denote the column space, row space, nullspace and left nullspace, respectively. \\

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

  \begin{exercise} % HW 1, Problem 1
    For each of the following part, construct a matrix \(\A\) with the specific properties specified. In this problem, \(n\times 1\) column vectors are written as ordered \(n\)-tuples.
    \begin{enumerate}
    \item The column space contains \((1,1,1)\) and the nullspace is the
      line of multiples of \((1,1,1,1)\).
    \item \(\A\) is a \(2\times 2\) matrix whose column space equals its
      nullspace.
    \item The nullspace of \(\A\) consists of all linear combinations of
      \((2,2,1,0)\) and \((3,1,0,1)\). [\emph{Hint}: Try a \(2\times 4\)
      matrix, and think of the ``column picture.'']
    \item The nullspace of \(\A\) consists of all multiples of
      \((4,3,2,1)\). 
    \item The column space contains \((1,1,5)\) and \((0,3,1)\) and the
      nullspace contains \((1,1,2)\). 
    \end{enumerate}
    
  \end{exercise}
  
  %\begin{solution} 
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  %\AImajor
  %\AIminor
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  %
  % To do.
  %\end{solution}

\medskip

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\begin{exercise} % HW 1, Problem 2
  Suppose we have a system \(\A\x=\b\) with particular solution \(\x_p=(2,4,0)\) and whose ``homogeneous'' solution \(\x_n\) (i.e., the nullspace of \(A\)) is the set of scalar multiples of \((1,1,1)\).
  \begin{enumerate}
  \item Construct such a \(2\times 3\) system. Sketch the ``grid picture.''
  \item Why can't there be a \(1\times 3\) system satisfying these
    conditions? Sketch the ``grid picture'' and show how it fails. 
  \end{enumerate}
\end{exercise}

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

\medskip

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

\begin{exercise} % HW 1, Problem 3
  Let \(\A\) be an \(m\times n\) matrix. 
  \begin{enumerate}
  \item Prove that the nullspace of \(\A\) is orthogonal to the row space. 
  \item Prove that the column space of \(\A\) is orthogonal to the left
    nullspace.
  \end{enumerate}
\end{exercise}

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