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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due September 19 (Wednesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 3.1, 3.2. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
3.1, \#15, 16, 23, 25--27, 29. Section 3.2, \#3, 6, 9, 10--18, 20,
28--31.  \\

\begin{enumerate}

  % Strang: Section 3.1, #10
\item For each subsets of $\R^3$, determine if it is a subspace. If
  not, give an explicit example of how it fails.
  \begin{enumerate}
  \item The plane of vectors
    $\b=(b_1,b_2,b_3)$ with $b_1=b_2$.
  \item The plane of vectors with $b_1=1$.
  \item The vectors with $b_1b_2b_3=0$.
  \item All linear combinations of
    $\v=(1,4,0)$ and $\w=(2,2,2)$.
  \item All vectors that satisfy $b_1+b_2+b_3=0$.
  \item All vectors with $b_1\leq b_2\leq b_3$.
  \end{enumerate}



  % Strang: Section 3.1, #12-13 (modified nearly beyond recognition)
\item Let $P$ be the plane in $\R^3$ with equation $x+y-2z=4$, which
  does not contain the zero-vector, and let $P_0$ be the parallel
  plane that passes through the origin.
  \begin{enumerate}
  \item Write the plane $P_0$ as an equation $ax+by+cz=d$.
  \item Find two vectors in $P$ and check that their sum is not in
    $P$.
  \item Find two non-colinear vectors in $P_0$ and check that their
    sum is in $P_0$.
  \item Write the set of vectors in $P_0$ as a linear combination
    $c\v+d\w$ for some $\v$ and $\w$.
  \item Write the set of vectors in $P$ as $\x_p+c\v+d\w$. That is,
    find such an $\x_p$ that works.
  \item Find a $3\times 2$ matrix $\A$ and a vector $\b$ such that the
    column space of $\A$ is the plane $P_0$ and the solutions to
    $\A\x=\b$ is the plane $P$. [\emph{Hint}: Any $\x_p$ that solves
      $\A\x=\b$ will work!]
  \end{enumerate}



  % Strang: Section 3.1, #4-5 (modified)
\item The matrix $\A=\begin{bmatrix}2&-2\\2&-2\end{bmatrix}$ can be
  thought of as a ``vector'' is the space $\M$ of all $2\times 2$
  matrices.
  \begin{enumerate}
  \item Write down the zero vector in this space, the vector
    $\frac{1}{2}\A$, and the vector $-\A$.
  \item What matrices are in the smallest subspace containing $\A$?
  \item Describe a subspace of $M$ that contains
    $\B=\begin{bmatrix}1&0\\0&0\end{bmatrix}$ but not
    $\C=\begin{bmatrix}0&0\\0&-1\end{bmatrix}$.
  \item If a subspace of $\M$ contains $\B$ and $\C$, must it contain
    $\I$?
  \item Describe a nontrivial subspace of $\M$ that contains no
    nonzero diagonal matrices.
  \end{enumerate}  



  % Strang: Section 3.1, # 17-18 (modified)
\item Fix $n$, and let $\M$ be the space of all $n\times n$
  matrices. For either of the following subsets of $\M$, determine
  whether it is a subspace. If true, give a reason, and if false, show
  why it fails with an explicit example.
  \begin{enumerate}
  \item The set of invertible matrices.
  \item The set of singular (non-invertible) matrices.
  \item The set of symmetric matrices ($\A^T=\A$).
  \item The set of skew-symmetric matrices ($\A^T=-\A$).
  \item The set of unsymmetric matrices ($\A^T\neq\A$).
  \end{enumerate}




  % Strang: Section 3.1, # 20 (modified)
\item For each system, determine which right sides (find a condition
  on $b_1$, $b_2$, and $b_3$) that makes them solvable. Describe the
  column space geometrically (line, plane, $\R^3$, etc.) in each case.
  \[
  \begin{bmatrix}1&4&2\\2&8&4\\-1&-4&-2\end{bmatrix}
  \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}
  =\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}\,\qquad\qquad\qquad
  \begin{bmatrix}1&4\\2&9\\-1&-4\end{bmatrix}
  \begin{bmatrix}x_1\\x_2\end{bmatrix}
  =\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}
  \]



  % Strang: Section 3.2, # 1-4 (modified)
\item Consider the following two matrices
  \[
  \A=\begin{bmatrix}1&2&2&4&6\\1&2&3&6&9\\0&0&1&2&3\end{bmatrix}
  \qquad\qquad\B=\begin{bmatrix}2&4&2\\0&4&4\\0&8&8\end{bmatrix}
  \]
  For each of these matrices, carry out the steps outlined below.
  \begin{enumerate}
  \item Reduce the matrix to its ordinary echelon form $\U$. Determine
    which are the free variables and which are the pivot
    variables. What is its rank?
  \item Find a \emph{special solution} for each free variable. That
    is, set the free variable to $1$ and the other free variables to
    $0$.
  \item By combining the special solutions, describe every solution to
    $\A\x=\0$ and $\B\x=\0$. This is the \emph{nullspace} of the
    matrix.
  \item Write down the \emph{nullspace matrix} $\N$, whose columns are
    the special solutions.
  \item By further row operations on each $\U$, find the reduced row
    echelon form $\bold{R}$. 
  \item Embedded within the nonzero rows of $\R$ are two (not
    necessarily contiguous) submatrices, one an identity matrix and
    the other a matrix $\F$. Identify $\F$. Now, $\N$ consists of two
    (not necessarily contiguous) submatrices: what are they?
  \item True or false: The nullspace of $\bold{R}$ equals the nullspace of
    $\U$?
  \end{enumerate}



  % Strang: Section 3.2 #25, 26, 21, 22, 23 (modified)
\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}: Let these be the
    ``special solutions'' of a matrix $\U$ where $x_3$ and $x_4$ are
    the free varibles.]
  \item The nullspace of $\A$ consists of all multiples of
    $(4,3,2,1)$. [\emph{Hint}: Let these be the special solutions
    of a matrix $\U$ where $x_4$ is the only free varible.]
  \item The column space contains $(1,1,5)$ and $(0,3,1)$ and the
    nullspace contains $(1,1,2)$.
  \end{enumerate}

\end{enumerate}

\end{document}
