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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
slide1

The Effects of

Linear Transformations

on Two –dimensional Objects

slide3

Timmy Twospace

Meets

Mr. Matrix

Alan Kaylor Cline

(An ill-conceived attempt to introduce humor into learning)

slide4

Dedicated to the Students of the

Inaugural Math 340L-CS Class

at the University of Texas at Austin,

Fall, 2012

slide8

This is Eee-Juan :

just that green spot.

We write it

slide9

Here’s the other friend.

He is Eee-too:

just that pink spot.

We write it

slide12

In fact, knowing where Mr. Matrix sends

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

slide14

Mr. Matrix is telling

Eee-Juan to go to

slide16

Mr. Matrix is telling

Eee-too to go to

slide22

(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|

slide23

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

slide25

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

slide27

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

Eee-too.

slide30

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

slide31

But before that, notice that I am not symmetric:

one arm is raised

– the other arm isn’t.

slide33

So here we go.

First, Mr. Matrix is

the “identity matrix”.

Mr. Matrix as the identity

slide36

Yup.

No change whatsoever.

slide43

This is called a “scaling”.

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

slide51

... I get twice as big.

Same shape – just twice as big.

slide54

My x-component s have been doubled but my

y-components were left alone.

slide63

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.

slide64

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

slide65

Let’s go the other way: halve the

x-coordinate and double the y-coordinate.

The 2 and the ½ are switched.

slide69

Now let’s go back to the identity - but add a non-zero in the upper right.

The upper right is now 1/2.

slide71

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

x-coordinates.

slide73

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.

slide75

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

y-coordinates.

slide76

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

slide79

Look closer.

Look at the arm I have raised.

Is this really a rotation?

slide83

But other than that exactly the same:

no shrinking,

no stretching.

slide84

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?

slide89

This is a reflection.

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

slide90

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

slide92

This matrix performs a counterclockwise rotation of an angle q.

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

slide94

Let’s try this rotation for

q = p/10 or 18 degrees.

slide102

You get the idea.

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

slide105

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

Stochastic Matrix

slide108

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%

slide124

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

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