Linear Algebra (Math 220)
Assignment due Tuesday, February 12

1.  Reading

Read § 2.5 in Matthews.

2.  Exercises

1. Let R(s, t) be the function from ${\mathbf{R}}^2$ to ${\mathbf{R}}^3$ defined by $$R\left(s,t\right)=\left(s+2t,-2s-t,-2s+2t\right)\text{.}$$

   1. Find equation(s) that characterize the set $S$ of all points $\left(x,y,z\right)$ in ${\mathbf{R}}^3$ that arise as $R\left(s,t\right)$ for at least one pair $\left(s,t\right)$.

   2. What kind of subset of ${\mathbf{R}}^3$ is $S$ ?

2. Find the inverse of the matrix $$M=\left(\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right)\text{.}$$

3. Find the smallest integer $k\ge 1$ for which the matrix power $M^k$ is the identity matrix when $M$ is the matrix of the previous exercise.

4. Find a simple formula for the $k$-th matrix power of the matrix $$T=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right)\text{.}$$