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\lhead{{\sf Bridge course (Algebra)}: Wednesday July 2}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 3 2025}
\cfoot{}\rfoot{Written by M.~Macauley}
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\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{enumerate}

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

\item 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}

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

\item 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}

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

\item 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{enumerate}


\end{document}

  
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