The Effects of Linear Transformations o n Two –dimensional Objects

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The Effects of Linear Transformations o n Two –dimensional Objects. or. 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

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Presentation Transcript

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

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

In fact, knowing where Mr. Matrix sends

Eee-Juan and Eee-too actually tells us everything.

Mr. Matrix is telling

Eee-Juan to go to

Mr. Matrix is telling

Eee-too to go to

(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

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

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

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

Eee-too.

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.

So here we go.

First, Mr. Matrix is

the “identity matrix”.

Mr. Matrix as the identity

Yup.

No change whatsoever.

This is called a “scaling”.

Notice the constant ½ on the diagonal of Mr. Matrix.

... I get twice as big.

Same shape – just twice as big.

My x-component s have been doubled but my

y-components were left alone.

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.

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.

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

Look closer.

Look at the arm I have raised.

Is this really a rotation?

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 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

The last example had q = p/2 or 90 degrees

Let’s try this rotation for

q = p/10 or 18 degrees.

You get the idea.

If we call this matrix R, then the total effect is R7.

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

Stochastic Matrix

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

16%

B

A

96%

84%

4%

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