Linear Algebra
Math 220
Exercises due Tuesday, February 5

The exercises below pertain to the function: $$f\left(x_1,x_2,x_3\right)=M\left(\begin{array}{c}\hfill x_1\\ \hfill x_2\\ \hfill x_3\end{array}\right)$$ where $$M=\left(\begin{array}{ccc}\hfill 1& \hfill -2& \hfill -1\\ \hfill 5& \hfill 4& \hfill -3\\ \hfill -2& \hfill -3& \hfill 1\end{array}\right)\text{.}$$

1. Put the matrix $M$ in reduced row echelon form.

2. Use your result in the preceding exercise as an aid to finding the set of all points $\left(x_1,x_2,x_3\right)$ for which $f\left(x_1,x_2,x_3\right)=\left(0,0,0\right)$.

3. What word describes the type of geometric object in $3$-dimensional space that is represented by your last answer?

4. Can you characterize the set of all points $\left(y_1,y_2,y_3\right)$ that occur as $f\left(x_1,x_2,x_3\right)$ for one or more points $\left(x_1,x_2,x_3\right)$?