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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

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.




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.


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



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.



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 Matrix.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: segments and circles and this box around me.

one arm is raised

– the other arm isn’t.


Pay special attention to the two arms. segments and circles and this box around me.


So here we go. segments and circles and this box around me.

First, Mr. Matrix is

the “identity matrix”.

Mr. Matrix as the identity


… and he transforms me to … segments and circles and this box around me.


Yup. segments and circles and this box around me.

No change whatsoever.


Pretty boring. Right? segments and circles and this box around me.

Written as I




(back to blue) identity matrix.


I’ve been shrunk in half. identity matrix.


This is called a “scaling”. identity matrix.

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


Let’s change that constant to 2. identity matrix.

Written as 2 I



The two processes are “inverses” of each other. process undid what the first did.

(½ I)-1 = 2 I



... I get twice as big. process undid what the first did.

Same shape – just twice as big.



Can you see I’ve been stretched? process undid what the first did.


My x-component s have been doubled process undid what the first did.but my

y-components were left alone.


My head is no longer a circle but an ellipse. process undid what the first did.




My of the 1 and 2.y-component s have been doubled but x-components were left alone.


Again my of the 1 and 2.head is an ellipse.


and again the of the 1 and 2.box around me is still a rectangle – now twice as tall.


Back to normal. Now let’s double the of the 1 and 2.

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 half as tall – so the area is the same as before.

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.


This is called a “shear”. y-coordinates are added to get the new


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… x-coordinates are added to get the new

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? x-coordinates are added to get the new


Look closer. x-coordinates are added to get the new

Look at the arm I have raised.

Is this really a rotation?





But other than that exactly the same: image of my former self.

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. last form but notice the -1.



Quiz Time: Watch this - is it a reflection or a rotation? last form but notice the -1.

Two -1’s


This is a reflection. last form but notice the -1.

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? counterclockwise. Notice the same arm is raised and there is no mirror image..


This matrix performs counterclockwise. Notice the same arm is raised and there is no mirror image..a counterclockwise rotation of an angle q.

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



Let’s try this rotation for direction.

q = p/10 or 18 degrees.


… and again… direction.


… and again… direction.


… and again… direction.


… and again… direction.


You get the idea. direction.

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



Here’s a state diagram corresponding to this matrix movements between “states”.

16%

B

A

96%

84%

4%


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%


Applying Mr. Matrix over and over is a way of finding the probability of moving from state A to state B is .04, …

“steady state”.

16%

B

A

96%

84%

4%


But let’s see what happens probability of moving from state A to state B is .04, …when Mr. Matrix is applied over and over to me.


And let’s skip forward an infinite number of steps to … probability of moving from state A to state B is .04, …


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


Signing off. “eigenvectors” corresponding to “eigenvalue” 1.

Bye.


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