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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 14 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 3110 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 3.3, 3.4, 3.5. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
3.3, \#1, 8, 9, 11, 13, 16, 17--22. Section 3.4, \#7, 13--17, 22, 24,
25, 27, 33. Section 3.5, \#4, 9. \medskip


\begin{enumerate}

  % Strang: Section 3.3, #2 (modified)
\item Find the reduced row echelon forms $\R$ and the rank of these matrices:
  \begin{enumerate}
  \item The $3\times 4$ matrix with all entries equal to $2$.
  \item The $3\times 4$ matrix with $a_{ij}=i+j-1$.
  \item The $3\times 4$ matrix with $a_{ij}=(-1)^j$. 
  \item The transposes of each of the three matrices above. 
  \end{enumerate}
    


  % Strang: Section 3.3, #7 (modified)
\item What are the ``special solutions'' to $\R\x=\0$ for the matrices
  $\R$ shown below, and what is their rank?
  \[
  \R=\begin{bmatrix}1&0&2&4\\0&1&3&5\\0&0&0&0\end{bmatrix}\,,\qquad\qquad
  \R=\begin{bmatrix}1&0&2\\0&0&0\\0&0&0\end{bmatrix}\,.
  \]

  
  
  % Strang: Section 3.4, #1 (modified)
\item Consider the system $\A\x=\b$, where
  \[
  \A=\begin{bmatrix}2&4&6&4\\2&5&7&6\\2&3&5&2\end{bmatrix}\,,\qquad\qquad
  \b=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}
  =\begin{bmatrix}4\\3\\5\end{bmatrix}\,.
  \]
  Solve this system by carrying out the following steps.
  \begin{enumerate}
    \item Reduce $[\A\;\;\b]$ to $[\U\;\;\c]$, turning the system
      $\A\x=\b$ into an upper triangular one, $\U\x=\c$.
    \item Find condition(s) on $b_1$, $b_2$, and $b_3$ for $\A\x=\b$
      to have a solution. Each ``zero row'' of $U$ will give you a
      condition.
    \item Describe the \emph{column space} $C(\A)$ of $\A$ as a subspace of
      $\R^3$. Express it as a linear combination of a \emph{minimal}
      number of vectors.
    \item Describe the \emph{nullspace} $\x_n:=N(\A)$ and find a basis for it
      (the special solutions).
    \item Find any \emph{particular solution} to $\A\x=\b$ and then
      the \emph{general solution}, which will have the form
      $\x=\x_p+\x_n$.
    \item Reduce $[\U\;\;\c]$ to $[\R\;\;\d]$, where $\R$ is the
      row-reduced echelon form and $\d$ is a particular solution.
  \end{enumerate}



% Strang: Section 3.4, #10,11 (modified)
\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}



% Strang: Section 3.4, #8 (modified)
\item Consider the following $3\times 3$ matrices.
  \[
  \A=\begin{bmatrix}1&2&1\\2&6&3\\0&2&5\end{bmatrix}\,,\qquad\qquad
  \B=\begin{bmatrix}1&1&1\\1&2&4\\2&4&8\end{bmatrix}\,.
  \]
  Carry out the following steps for both $\A$ and $\B$.
  \begin{enumerate}
  \item Determine which vectors $(b_1,b_2,b_3)$ are in the column
    space. [\emph{Hint}: Each ``zero row'' should give you a
      condition on $b_1$, $b_2$, $b_3$.]
  \item What combination of the rows give the zero row?
  \item What is the relationship between Parts~(a) and (b)?
  \end{enumerate}



% Strang: Section 3.4, #31 (modified)
\item Find matrices $\A$ and $\B$ with the given property or explain
  why there is none. [\emph{Hint}: Recall that
    $\x=\x_p+\x_n$. Sketching the ``grid picture'' isn't necessary,
    but it may help.]
  \begin{enumerate}
  \item The only solution of
    $\A\x=\begin{bmatrix}1\\2\\3\end{bmatrix}$ is
    $\x=\begin{bmatrix}1\\0\end{bmatrix}$.
  \item The only solution of
    $\B\x=\begin{bmatrix}1\\0\end{bmatrix}$ is
    $\x=\begin{bmatrix}1\\2\\3\end{bmatrix}$.
  \end{enumerate}



% Strang: Section 3.5, #1 (modified)
\item Consider the following four vectors:
  \[
  \v_1=\begin{bmatrix}1\\0\\0\end{bmatrix}\,,\qquad
  \v_2=\begin{bmatrix}1\\1\\0\end{bmatrix}\,,\qquad
  \v_3=\begin{bmatrix}1\\1\\1\end{bmatrix}\,,\qquad
  \v_4=\begin{bmatrix}2\\3\\4\end{bmatrix}\,.
  \]
  \begin{enumerate}
  \item Show that $\v_1$, $\v_2$, $\v_3$ are linearly independent but
    $\v_1$, $\v_2$, $\v_3$, $\v_4$ are linearly dependent.
  \item Solve $c_1\v_1+c_2\v_2+c_3\v_3+c_4\v_4=\0$, or alternatively,
    $\A\x=\0$ where the $\v_i$'s are the columns of $\A$.
  \end{enumerate}



% Strang: Section 3.5, #10 (modified)
\item Let $P$ be the hyperplane $x+2y-3z-t=0$ in $\R^4$. 
  \begin{enumerate}
    \item Find two linearly independent vectors on $P$.
    \item Find three linearly independent vectors on $P$.
    \item Why can you not find four linearly independent vectors on $P$?
    \item Find a matrix $A$ whose column space is $P$, and a matrix
      $B$ whose nullspace is $P$.
  \end{enumerate}

\end{enumerate}

\end{document}
