Linear Algebra (Math 220)
Assignment due Thursday, February 7

1.  Reading

Read §§ 2.2 – 2.3 in Matthews.

2.  Exercises

1. Let $A$ be the $3\times 4$ matrix $$A=\left(\begin{array}{cccc}\hfill 2& \hfill 3& \hfill 1& \hfill -4\\ \hfill 3& \hfill -2& \hfill -1& \hfill 5\\ \hfill 5& \hfill 1& \hfill 0& \hfill 1\end{array}\right)\text{.}$$ Let $f$ be the function from ${\mathbf{R}}^4$ to ${\mathbf{R}}^3$ given by $f\left(x\right)=Ax$.

   1. Find all points $x$ in ${\mathbf{R}}^4$ for which $f\left(x\right)=0$.

   2. Find all points $x$ in ${\mathbf{R}}^4$ for which $$f\left(x\right)=\left(\begin{array}{c}\hfill 4\\ \hfill -1\\ \hfill 3\end{array}\right)\text{.}$$

   3. Characterize the set of points $y$ in ${\mathbf{R}}^3$ for which the relation $f\left(x\right)=y$ holds for at least one point $x$ in ${\mathbf{R}}^4$.

2. Let $M$ be the matrix $$\left(\begin{array}{ccc}\hfill 1& \hfill 5& \hfill -2\\ \hfill -2& \hfill 4& \hfill -3\\ \hfill -1& \hfill -3& \hfill 1\end{array}\right),$$ and let $g$ be the function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ given by $g\left(x\right)=Mx$.

   1. Find all points $x$ in ${\mathbf{R}}^3$ for which $g\left(x\right)=0$.

   2. Find all points $x$ in ${\mathbf{R}}^3$ for which $$g\left(x\right)=\left(\begin{array}{c}\hfill 1\\ \hfill -5\\ \hfill 3\end{array}\right)\text{.}$$

   3. Find all points $x$ in ${\mathbf{R}}^3$ for which $$g\left(x\right)=\left(\begin{array}{c}\hfill -1\\ \hfill 2\\ \hfill 1\end{array}\right)\text{.}$$

   4. Characterize the set of points $y$ in ${\mathbf{R}}^3$ for which the relation $g\left(x\right)=y$ holds for at least one point $x$ in ${\mathbf{R}}^3$.