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The Effects of Linear Transformations on Two dimensional Objects

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or

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Timmy Twospace Meets Mr. Matrix (An ill-conceived attempt to introduce humor into learning) Alan Kaylor Cline

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Dedicated to the Students of the Inaugural Math 340L-CS Class at the University of Texas at Austin, Fall, 2012

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

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I want you to meet two friends of mine: Eee-Juan and Eee-too.

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For the moment, I going to be invisible.

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We write it This is Eee-Juan : just that green spot.

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We write it Heres the other friend. He is Eee-too: just that pink spot.

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Mr. Matrix and this is Mr. Matrix.

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Mr. Matrix tells us where to go.

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

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Eee-Juan gets his instructions from the first column of Mr. Matrix

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Mr. Matrix is telling Eee-Juan to go to

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Eee-too gets his instructions from the second column of Mr. Matrix.

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Mr. Matrix is telling Eee-too to go to

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and those are enough instructions to tell where everything moves.

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

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

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And all of the points in this square

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are transformed to all of the points in this parallelogram

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|ad-bc| 1 (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

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

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And this even applies to me

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

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

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

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This is where Mr. Matrix sends my points.

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

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

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But before that, notice that I am not symmetric: one arm is raised the other arm isnt.

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Pay special attention to the two arms.

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So here we go. First, Mr. Matrix is the identity matrix. Mr. Matrix as the identity

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and he transforms me to

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Yup. No change whatsoever.

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Pretty boring. Right? Written as I

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

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and he transforms me to

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(back to blue)

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Ive been shrunk in half.

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This is called a scaling. Notice the constant on the diagonal of Mr. Matrix.

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Lets change that constant to 2. Written as 2 I

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

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The two processes are inverses of each other. ( I) -1 = 2 I

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and if we were to apply this scaling again to me

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... I get twice as big. Same shape just twice as big.

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

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Can you see Ive been stretched?

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

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My head is no longer a circle but an ellipse.

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

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

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

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Again my head is an ellipse.

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

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Back to normal. Now lets double the x-coordinate and halve the y-coordinate at the same time. Notice the 2 and the .

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

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Lets go the other way: halve the x-coordinate and double the y- coordinate. The 2 and the are switched.

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

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

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Now lets go back to the identity - but add a non- zero in the upper right. The upper right is now 1/2.

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

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This is called a shear.

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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.

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

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Moving on So what will this do? It looks sort of like the identity. The 1s and 0s are reversed from the identity

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Do you believe Ive been rotated?

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Look closer. Look at the arm I have raised. Is this really a rotation?

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

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

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

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But other than that exactly the same: no shrinking, no stretching.

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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?

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This is a rotation through 90 degrees.

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

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Quiz Time: Watch this - is it a reflection or a rotation? Two -1s

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

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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

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So what is a general rotation?

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This matrix performs a counterclockwise rotation of an angle The last example had = /2 or 90 degrees

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Moving counterclockwise is considered the positive direction.

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Lets try this rotation for = /10 or 18 degrees.

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and again

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You get the idea. If we call this matrix R, then the total effect is R 7.

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

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

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

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Heres a state diagram corresponding to this matrix A B 4% 84%96% 16%

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Thus, the probability of staying in state A is.96, the probability of moving from state A to state B is.04, A B 4% 84%96% 16%

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Applying Mr. Matrix over and over is a way of finding the steady state. A B 4% 84%96% 16%

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

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And lets skip forward an infinite number of steps to

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

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Timmy Twospace signing off. Bye.