10.10 Homework: using matrices for transformation

Harpoona’s best friend is a fish named Sam, whose initial position is represented by the matrix:

$\left[S_1\right]=\left[\begin{array}{cccccc}0& 4& 4& 0& 0& 4\\ 0& 0& 3& 3& 0& 3\end{array}\right]$

Draw Sam:

When the matrix $T=\frac{1}{2}\left[\begin{array}{cc}\sqrt{3}& -1\\ 1& \sqrt{3}\end{array}\right]$ is multiplied by any matrix, it effects a powerful transformation on that matrix. Below, write the matrix $S_2=T{S}_1$ . (You may use 1.7 as an approximation for $\sqrt{3}$ .)

Draw Sam’s resulting condition, $S_2$ .

The matrix $T^{\mathrm{–1}}$ will, of course, do the opposite of $T$ . Find $T^{\mathrm{–1}}$ . (You can use the formula for the inverse matrix that we derived in class, instead of starting from first principles. But make sure to first multiply the $\frac{1}{2}$ into $T$ , so you know what the four elements are!)

Sam now undergoes this transformation, so his new state is given by $S_3=T^{\mathrm{–1}}{S}_2$ . Find $S_3$ and graph his new position.

Finally, Sam goes through $T^{\mathrm{–1}}$ again, so his final position is $S_4=T^{\mathrm{–1}}{S}_3$ . Find and graph his final position.

Describe in words: what do the transformations $T$ and $T^{\mathrm{–1}}$ do, in general, to any shape?