The Effects of Linear Transformations o n Two –dimensional Objects

The Effects of Linear Transformations on Two –dimensional Objects

Timmy Twospace Meets Mr. Matrix Alan Kaylor Cline (An ill-conceived attempt to introduce humor into learning)

Dedicated to the Students of the Inaugural Math 340L-CS Class at the University of Texas at Austin, Fall, 2012

Hi. I’m Timmy Twospace and I want to show you what happens to me when Mr. Matrix does his thing.

I want you to meet two friends of mine: Eee-Juan and Eee-too.

For the moment, I going to be invisible.

This is Eee-Juan : just that green spot. We write it

Here’s the other friend. He is Eee-too: just that pink spot. We write it

… and this is Mr. Matrix. Mr. Matrix

Mr. Matrix tells us where to go.

In fact, knowing where Mr. Matrix sends Eee-Juan and Eee-too actually tells us everything.

Eee-Juan gets his instructions from the first column of Mr. Matrix

Mr. Matrix is telling Eee-Juan to go to

Eee-too gets his instructions from the second column of Mr. Matrix.

Mr. Matrix is telling Eee-too to go to

… and those are enough instructions to tell where everything moves.

For example, this blue point is half of Eee-Juan plus twice Eee-too.

So the point moves to twice where Eee-Juan moves plus one half of where Eee-too moves.

And all of the points in this square …

are transformed to all of the points in this parallelogram

(and by the way, the area of the parallelogram is |ad-bc| times the area of the square.) ad-bc is the “determinant” of this matrix 1 |ad-bc|

Once again, knowing where Mr. Matrix sends Eee-Juan and Eee-too actually tells us everything.

And this even applies to me

First realize that, amusing as I am, I‘m actually just some points in the plane: line segments and circles.

So, all of my points move under the instructions of Mr. Matrix.

Every one of my points is just a sum of some amount of Eee-Juan and some amount of Eee-too.

This is where Mr. Matrix sends my points.

We are going to see what happens to me with various versions of Mr. Matrix.

You should pay attention to what happens to my line segments and circles and this box around me.

But before that, notice that I am not symmetric: one arm is raised – the other arm isn’t.

Pay special attention to the two arms.

So here we go. First, Mr. Matrix is the “identity matrix”. Mr. Matrix as the identity

… and he transforms me to …

Yup. No change whatsoever.

Pretty boring. Right? Written as I

This time Mr. Matrix is just half of what he was as the identity matrix. Written as ½ I

…and he transforms me to…

(back to blue)

I’ve been shrunk in half.

This is called a “scaling”. Notice the constant ½ on the diagonal of Mr. Matrix.

Let’s change that constant to 2. Written as 2 I

And now I am back to my original self. Notice the second process undid what the first did.

The two processes are “inverses” of each other. (½ I)-1 = 2 I

… and if we were to apply this scaling again to me…

The Effects of Linear Transformations o n Two –dimensional Objects