Linear Algebra
Math 220
Assignment due Thursday, January 31

1. Solve for $x$, $y$, and $z$ in terms of $u$, $v$, and $w$. $$\begin{array}{cccc}& \hfill x-y+z& \hfill =\hfill & u\hfill \\ & \hfill 5x-4y+3z& \hfill =\hfill & v\hfill \\ & \hfill 3x-3y+2z& \hfill =\hfill & w\hfill \end{array}$$

2. For given constants $a$, $b$, $c$, and $d$ solve the following system of linear equations for $x$ and $y$ in terms of $u$ and $v$. $$\begin{array}{cccc}& \hfill ax+by& \hfill =\hfill & u\hfill \\ & \hfill cx+dy& \hfill =\hfill & v\hfill \end{array}$$

3. Let $M$ be the matrix $$M=\left(\begin{array}{ccc}\hfill 1& \hfill -1& \hfill 1\\ \hfill 5& \hfill -4& \hfill 3\\ \hfill 3& \hfill -3& \hfill 2\end{array}\right)\text{.}$$ Solve the system of linear equations $$M\left(\begin{array}{c}\hfill x\\ \hfill y\\ \hfill z\end{array}\right)=b$$ when $b$ is: $$\text{(a)}\left(\begin{array}{c}\hfill 1\\ \hfill 0\\ \hfill 0\end{array}\right)\text{(b)}\left(\begin{array}{c}\hfill 0\\ \hfill 1\\ \hfill 0\end{array}\right)\text{(c)}\left(\begin{array}{c}\hfill 0\\ \hfill 0\\ \hfill 1\end{array}\right)\text{(d)}\left(\begin{array}{c}\hfill 2\\ \hfill -3\\ \hfill 1\end{array}\right)\text{.}$$ Suggestion: Review the solution of the first exercise on the last assignment.

4. Let $N$ be the matrix $$N=\left(\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 1\\ \hfill 5& \hfill -4& \hfill 3\\ \hfill 3& \hfill -3& \hfill 2\end{array}\right)\text{.}$$ Find all solutions of the system of linear equations $$N\left(\begin{array}{c}\hfill x\\ \hfill y\\ \hfill z\end{array}\right)=b$$ when $b$ is: $$\text{(a)}\left(\begin{array}{c}\hfill 1\\ \hfill 0\\ \hfill 0\end{array}\right)\text{(b)}\left(\begin{array}{c}\hfill 0\\ \hfill 1\\ \hfill 0\end{array}\right)\text{(c)}\left(\begin{array}{c}\hfill 1\\ \hfill 1\\ \hfill 1\end{array}\right)\text{.}$$ Note: Things become very different with the change of a single matrix entry between the matrix $M$ of the first exercise and the present matrix $N$.