10.9 Using matrices for transformation

Explain, in words, how this matrix represents her position. That is, how can this matrix give instructions to a computer on exactly how to draw Harpoona?

The transformation $\frac{1}{2}\left[H\right]$ is applied to Harpoona.

• A

  Write down the resulting matrix.

• B

  Draw Harpoona after this transformation.

• C

  Then answer this question in words: in general , what does the transformation $\frac{1}{2}\left[H\right]$ do to a picture?

Now, Harpoona is going to swim three units to the left. Write below a general transformation that can be applied to any 2×4 matrix to move a drawing three units to the left.

Harpoona—in her original configuration before she was transformed in either way—now undergoes the transformation $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ $\left[H\right]$ .

• A

  Write the new matrix that represents Harpoona.

• B

  Draw Harpoona after this transformation.

• C

  In the space below, answer this question in words: in general , what does the transformation $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ $\left[H\right]$ do to a picture?

Now: in the movie’s key scene, the audience is looking down from above on Harpoona who begins the scene happily floating on the surface of the water. As the scene progresses, our heroine spins around and around in a whirlpool as she is slowly being sucked down to the bottom of the sea. “Being sucked down” is represented visually, of course, by shrinking.

• A

  Write a single transformation that will rotate Harpoona by 900 and shrink her.

• B

  Apply this transformation four times to Harpoona’s original state, and compute the resulting matrices that represent her next four states.

• C

  Now draw all four states—preferably in different colors or something.