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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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The Effects of Linear Transformations o n Two –dimensional Objects

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The effects of linear transformations o n two dimensional objects

The Effects of

Linear Transformations

on Two dimensional Objects


The effects of linear transformations o n two dimensional objects

or


The effects of linear transformations o n two dimensional objects

Timmy Twospace

Meets

Mr. Matrix

Alan Kaylor Cline

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


The effects of linear transformations o n two dimensional objects

Dedicated to the Students of the

Inaugural Math 340L-CS Class

at the University of Texas at Austin,

Fall, 2012


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

I want you to meet two friends of mine:

Eee-Juan and Eee-too.


The effects of linear transformations o n two dimensional objects

For the moment, I going to be invisible.


The effects of linear transformations o n two dimensional objects

This is Eee-Juan :

just that green spot.

We write it


The effects of linear transformations o n two dimensional objects

Heres the other friend.

He is Eee-too:

just that pink spot.

We write it


The effects of linear transformations o n two dimensional objects

and this is Mr. Matrix.

Mr. Matrix


The effects of linear transformations o n two dimensional objects

Mr. Matrix tells us where to go.


The effects of linear transformations o n two dimensional objects

In fact, knowing where Mr. Matrix sends

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


The effects of linear transformations o n two dimensional objects

Eee-Juan gets his instructions from the first column of

Mr. Matrix


The effects of linear transformations o n two dimensional objects

Mr. Matrix is telling

Eee-Juan to go to


The effects of linear transformations o n two dimensional objects

Eee-too gets his instructions from the second column of

Mr. Matrix.


The effects of linear transformations o n two dimensional objects

Mr. Matrix is telling

Eee-too to go to


The effects of linear transformations o n two dimensional objects

and those are enough instructions to tell where everything moves.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

And all of the points in this square


The effects of linear transformations o n two dimensional objects

are transformed to all of the points in this parallelogram


The effects of linear transformations o n two dimensional objects

(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|


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

And this even applies

to me


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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

Eee-too.


The effects of linear transformations o n two dimensional objects

This is what happens to my points.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

But before that, notice that I am not symmetric:

one arm is raised

the other arm isnt.


The effects of linear transformations o n two dimensional objects

Pay special attention to the two arms.


The effects of linear transformations o n two dimensional objects

So here we go.

First, Mr. Matrix is

the identity matrix.

Mr. Matrix as the identity


The effects of linear transformations o n two dimensional objects

and he transforms me to


The effects of linear transformations o n two dimensional objects

Yup.

No change whatsoever.


The effects of linear transformations o n two dimensional objects

Pretty boring. Right?

Written as I


The effects of linear transformations o n two dimensional objects

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

Written as I


The effects of linear transformations o n two dimensional objects

and he transforms me to


The effects of linear transformations o n two dimensional objects

(back to blue)


The effects of linear transformations o n two dimensional objects

Ive been shrunk in half.


The effects of linear transformations o n two dimensional objects

This is called a scaling.

Notice the constant on the diagonal of Mr. Matrix.


The effects of linear transformations o n two dimensional objects

Lets change that constant to 2.

Written as 2 I


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

The two processes are inverses of each other.

( I)-1 = 2 I


The effects of linear transformations o n two dimensional objects

and if we were to apply this scaling again to me


The effects of linear transformations o n two dimensional objects

... I get twice as big.

Same shape just twice as big.


The effects of linear transformations o n two dimensional objects

Now lets see what this one does with one 2 and one 1.


The effects of linear transformations o n two dimensional objects

Can you see Ive been stretched?


The effects of linear transformations o n two dimensional objects

My x-component s have been doubled but my

y-components were left alone.


The effects of linear transformations o n two dimensional objects

My head is no longer a circle but an ellipse.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Im back to regular and now well reverse the positions of the 1 and 2.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Again my head is an ellipse.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Back to normal. Now lets double the

x-coordinate and halve the y-coordinate at the same time.

Notice the 2 and the .


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Lets go the other way: halve the

x-coordinate and double the y-coordinate.

The 2 and the are switched.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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

The upper right is now 1/2.


The effects of linear transformations o n two dimensional objects

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

x-coordinates.


The effects of linear transformations o n two dimensional objects

This is called a shear.


The effects of linear transformations o n two dimensional objects

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 effects of linear transformations o n two dimensional objects

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

y-coordinates.


The effects of linear transformations o n two dimensional objects

Moving on

So what will this do?

It looks sort of like the identity.

The 1s and 0s are reversed from the identity


The effects of linear transformations o n two dimensional objects

Do you believe Ive been rotated?


The effects of linear transformations o n two dimensional objects

Look closer.

Look at the arm I have raised.

Is this really a rotation?


The effects of linear transformations o n two dimensional objects

Nope. Its a reflection. My x- and y-components have been reversed.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

But other than that exactly the same:

no shrinking,

no stretching.


The effects of linear transformations o n two dimensional objects

Im back to normal and Mr. Matrix is very similar to his last form but notice the -1.

See the -1 in the lower left?


The effects of linear transformations o n two dimensional objects

This is a rotation through 90 degrees.


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Quiz Time: Watch this - is it a reflection or a rotation?

Two -1s


The effects of linear transformations o n two dimensional objects

This is a reflection.

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


The effects of linear transformations o n two dimensional objects

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 effects of linear transformations o n two dimensional objects

So what is a general rotation?


The effects of linear transformations o n two dimensional objects

This matrix performs a counterclockwise rotation of an angle q.

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


The effects of linear transformations o n two dimensional objects

Moving counterclockwise is considered the positive direction.


The effects of linear transformations o n two dimensional objects

Lets try this rotation for

q = p/10 or 18 degrees.


The effects of linear transformations o n two dimensional objects

and again


The effects of linear transformations o n two dimensional objects

and again


The effects of linear transformations o n two dimensional objects

and again


The effects of linear transformations o n two dimensional objects

and again


The effects of linear transformations o n two dimensional objects

You get the idea.

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


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

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

Stochastic Matrix


The effects of linear transformations o n two dimensional objects

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


The effects of linear transformations o n two dimensional objects

Heres a state diagram corresponding to this matrix

16%

B

A

96%

84%

4%


The effects of linear transformations o n two dimensional objects

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%


The effects of linear transformations o n two dimensional objects

Applying Mr. Matrix over and over is a way of finding the

steady state.

16%

B

A

96%

84%

4%


The effects of linear transformations o n two dimensional objects

But lets see what happens when Mr. Matrix is applied over and over to me.


The effects of linear transformations o n two dimensional objects

And lets skip forward an infinite number of steps to


The effects of linear transformations o n two dimensional objects

And now Im fixed. All of my points are called eigenvectors corresponding to eigenvalue 1.


The effects of linear transformations o n two dimensional objects

Signing off.

Bye.


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