Exercise 1

Suppose that $T$ is the matrix transformation defined by the matrix $A$ and $S$ is the matrix transformation defined by $B$ where $$A = \left[\begin{array}{rrr} 3 & -1 & 0 \\ 1 & 2 & 2 \\ -1 & 3 & 2 \\ \end{array}\right], \qquad B = \left[\begin{array}{rrr} 1 & -1 & 0 \\ 2 & 1 & 2 \\ \end{array}\right]$$.

1. If $T:\real^n\to\real^m$, what are the values of $m$ and $n$? What values of $m$ and $n$ are appropriate for the transformation $S$?

2. Evaluate $T\left(\threevec{1}{-3}{2}\right)$.

3. Evaluate $S\left(\threevec{-2}{2}{1}\right)$.

4. Evaluate $S\circ T\left(\threevec{1}{-3}{2}\right)$.

5. Find the matrix $C$ that defines the matrix transformation $S\circ T$.