The Effects of Linear Transformations on Two –dimensional Objects

or

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

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.

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

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

Mr. Matrix … and this is 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

|ad-bc| 1 (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

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…

... I get twice as big. Same shape – just twice as big.

Now let’s see what this one does with one 2 and one 1.

Can you see I’ve been stretched?

My x-component s have been doubled but my y-components were left alone.

My head is no longer a circle but an ellipse.

The box around me is still a rectangle – just twice as wide.

I’m back to regular and now we’ll reverse the positions of the 1 and 2.

My y-component s have been doubled but x- components were left alone.

Again my head is an ellipse.

and again the box around me is still a rectangle – now twice as tall.

Back to normal. Now let’s double the x-coordinate and halve the y-coordinate at the same time. Notice the 2 and the ½.

Big time squishing, right? The box is twice as wide and half as tall – so the area is the same as before.

Let’s go the other way: halve the x-coordinate and double the y- coordinate. The 2 and the ½ are switched.

Those transformations stretched or shrank the x- or y-coordinate – or both.

Mr. Matrix was “diagonal”: non-zeros only in the upper left and lower right positions.

Now let’s go back to the identity - but add a non- zero in the upper right. The upper right is now 1/2.

The y-coordinates are left alone. The x- plus one half the y-coordinates are added to get the new x-coordinates.

This is called a “shear”.

There is another shear: We go back to the identity but add a non- zero in the lower left. The lower left is now 1/2.

The x-coordinates are left alone. The y- plus one half the x-coordinates are added to get the new y-coordinates.

Moving on… So what will this do? It looks sort of like the identity. The 1’s and 0’s are reversed from the identity

Do you believe I’ve been rotated?

Look closer. Look at the arm I have raised. Is this really a rotation?

Nope. It’s a “reflection”. My x- and y- components have been reversed.

This is easier to see if I draw in this 45 degree line.

A reflection is a flipping across some line. I am a mirror image of my former self.

But other than that exactly the same: no shrinking, no stretching.

I’m back to normal and Mr. Matrix is very similar to his last form but notice the -1. See the -1 in the lower left?

This is a rotation through 90 degrees.

Notice it is not a reflection - not a mirror image.

Quiz Time: Watch this - is it a reflection or a rotation? Two -1’s

This is a reflection. Do you see that it is a mirror image across the line?

On the other hand, this one is a rotation of 90 degrees counterclockwise. Notice the same arm is raised and there is no mirror image.. One -1

So what is a general rotation?

This matrix performs a counterclockwise rotation of an angle  The last example had  =  /2 or 90 degrees

Moving counterclockwise is considered the “positive” direction.

Let’s try this rotation for  =  /10 or 18 degrees.

… and again…

You get the idea. If we call this matrix R, then the total effect is R 7.

Finally, we will see what happens when Mr. Matrix transforms me over and over.

This is a special matrix called a “stochastic matrix”: no negative numbers and each column has a sum of 1. Stochastic Matrix

It is sometimes used to describe the probabilities of movements between “states”.

Here’s a state diagram corresponding to this matrix A B 4% 84%96% 16%

Thus, the probability of staying in state A is.96, the probability of moving from state A to state B is.04, … A B 4% 84%96% 16%

Applying Mr. Matrix over and over is a way of finding the “steady state”. A B 4% 84%96% 16%

But let’s see what happens when Mr. Matrix is applied over and over to me.

And let’s skip forward an infinite number of steps to …

And now I’m fixed. All of my points are called “eigenvectors” corresponding to “eigenvalue” 1.

Timmy Twospace signing off. Bye.