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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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on Two –dimensional Objects
Alan Kaylor Cline
(An ill-conceived attempt to introduce humor into learning)
Inaugural Math 340L-CS Class
at the University of Texas at Austin,
Hi. I’m Timmy Twospace and I want to show you what happens to me when Mr. Matrix does his thing.
Eee-Juan and Eee-too.
just that green spot.
We write it
He is Eee-too:
just that pink spot.
We write it
Eee-Juan and Eee-too actually tells us everything.
Eee-Juan to go to
Eee-too to go to
… and those are enough instructions to tell where everything moves.
So the point moves to twice where Eee-Juan moves plus one half of where Eee-too moves.
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.
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
We are going to see what happens to me with various versions on Mr. Matrix.
You should pay attention to what happens to my line segments and circles and this box around me.
one arm is raised
– the other arm isn’t.
First, Mr. Matrix is
the “identity matrix”.
Mr. Matrix as the identity
No change whatsoever.
Written as I
This time Mr. Matrix is just half of what he was as the identity matrix.
Written as ½ I
Notice the constant ½ on the diagonal of Mr. Matrix.
Written as 2 I
And now I am back to my original self. Notice the second process undid what the first did.
(½ I)-1 = 2 I
Same shape – just twice as big.
y-components were left alone.
I’m back to regular and now we’ll reverse the positions of the 1 and 2.
and again the box around me is still a rectangle – now twice as tall.
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.
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
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
So what will this do?
It looks sort of like the identity.
The 1’s and 0’s are reversed from the identity
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.
A reflection is a flipping across some line. I am a mirror image of my former self.
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?
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..
The last example had q = p/2 or 90 degrees
q = p/10 or 18 degrees.
If we call this matrix R, then the total effect is R7.
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.
It is sometimes used to describe the probabilities of movements between “states”.
Thus, the probability of staying in state A is .96, the probability of moving from state A to state B is .04, …
But let’s see what happens when Mr. Matrix is applied over and over to me.
And now I’m fixed. All of my points are called “eigenvectors” corresponding to “eigenvalue” 1.