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\noindent
{\Large Math 211 \qquad Homework 7 \hfill Sample Texfile}

\medskip
\noindent{\bf 1.}
Use Cramer's rule to compute the solution of the
system
$$-5x_1+3x_2=9, \qquad
3x_1-x_2=-5.$$

\medskip
\noindent{\bf 2.}
Use Cramer's rule to compute the solution of the
system. 
$$2x_1+x_2+x_3=4, \qquad -x_1+3x_3=2, \qquad 3x_1+x_2+3x_3=2.$$



\medskip
\noindent{\bf 3.}
Calculate the adjugate (adjoint)  and the 
inverse of the following matrix:
$$A = \left[\matrix{0&-2&-1\cr 3&0&0\cr -1&1&1}\right].$$


\medskip\noindent
{\bf 4.}
Compute the area of parallelogram with the vertexes: 
$(0,-2), (6,-1), (-3,1), (3,2)$.

[Hint: Add $(0,2)$ to all the points to translate the parallelogram with $(0,0)$ as
a vertex.]  

\medskip
\noindent{\bf 5.}
Let
$$W = \left\{\left[\matrix{x\cr y\cr}\right]: xy \geq 0 \right\}.$$

\begin{enumerate}
 \item Show that $c{\bf u} \in W$  for any ${\bf u} \in W$ and any $c\in \mathbb{R}$.
 

\item Find two vectors in $W$ whose sum is not in $W$.

[Hint: for the second part, you need to find $u = \left[\matrix{x_1\cr y_1\cr}\right]$
such that $x_1 y_1 \ge 0$ and $v = \left[\matrix{x_2\cr y_2\cr}\right]$ so that 
$x_2y_2 \ge 0$. But $u + y$ have entries of different signs.]
\end{enumerate}

\medskip
\noindent{\bf 6.}
Let $W$ be all the vectors of the form $\left[\matrix{2s+4t\cr 2s\cr 2s-3t
\cr 5t}\right]$. Show that $W$ is a subspace of $\mathbb{R}^4$. 

[Hint: Note that you need only show that the following three rules hold for $W$. 
(a) $0 \in W$, 
(b) $u+v \in W$ if $u, v \in W$, and (c) $cu \in W$ if $c \in \IR$ and $u \in W$.]


\medskip
\noindent{\bf 7.}
Let $W = \left\{ \left[\matrix{x_1\cr x_2\cr x_3\cr}\right]: x_1 + x_2 + x_3 = 1\right\}$.
Show that $W$ is not a vector space.

[Hint: You need to show that one of the three rules (a)--(c) fails in the previous hint.]


\medskip
\noindent{\bf 8.}
Let $F$ be a fixed $3 \times 2$ matrix, and let $W = \{A: A \hbox{ is } 2\times 4, FA = 0\}$.
Show that $W$ is a vector space.

[Hint: (a) Show that $A = O_{2\times 4}$ lies in $W$. (b) Show that if $X, Y$ are $2\times 4$ 
matrices in $W$, then $X+Y$ is a matrix in $W$. (c) Show that if $c \in \IR$ and $X \in W$,
then $cX \in W$.]

\medskip\noindent
{\bf 9.} 
Let  $H$ and $K$ be subspaces of a vector space $V$. 
Show that $H+K = \{x+y: x \in H, y \in K\}$ is a subspace.

[Hint: Note that $H, K$ satisfy (a), (b), (c) rules in the Hints of Problem 6.
Now, to show $H+K$ also satisfies the rules, one has to check:

(a) $0$ can be written as $h+k$ with $h \in H, k \in K$ so that $0 \in H+K$.

(b) If $u = h_1+k_1, v = h_2 + k_2 \in H+K$ where $h_1, h_2 \in H, k_1, k_2 \in K$,
then $u+v$ can be written as $h+k$ with $h \in H$ and $k \in K$.

(c) If $c \in \IR$ and $u = h_1 + k_1\in H+K$ with $h_1 \in H, k_1 \in K$, 
then $cu = h+k$ for some $h \in H$ and $k \in K$.]
 

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