Transcript of The Effects of Linear Transformations on Two –dimensional Objects.
- Slide 1
- The Effects of Linear Transformations on Two dimensional
Objects
- Slide 2
- or
- Slide 3
- Timmy Twospace Meets Mr. Matrix (An ill-conceived attempt to
introduce humor into learning) Alan Kaylor Cline
- Slide 4
- Dedicated to the Students of the Inaugural Math 340L-CS Class
at the University of Texas at Austin, Fall, 2012
- Slide 5
- Hi. Im Timmy Twospace and I want to show you what happens to me
when Mr. Matrix does his thing.
- Slide 6
- I want you to meet two friends of mine: Eee-Juan and
Eee-too.
- Slide 7
- For the moment, I going to be invisible.
- Slide 8
- We write it This is Eee-Juan : just that green spot.
- Slide 9
- We write it Heres the other friend. He is Eee-too: just that
pink spot.
- Slide 10
- Mr. Matrix and this is Mr. Matrix.
- Slide 11
- Mr. Matrix tells us where to go.
- Slide 12
- In fact, knowing where Mr. Matrix sends Eee-Juan and Eee-too
actually tells us everything.
- Slide 13
- Eee-Juan gets his instructions from the first column of Mr.
Matrix
- Slide 14
- Mr. Matrix is telling Eee-Juan to go to
- Slide 15
- Eee-too gets his instructions from the second column of Mr.
Matrix.
- Slide 16
- Mr. Matrix is telling Eee-too to go to
- Slide 17
- and those are enough instructions to tell where everything
moves.
- Slide 18
- For example, this blue point is half of Eee-Juan plus twice
Eee-too.
- Slide 19
- So the point moves to twice where Eee-Juan moves plus one half
of where Eee-too moves.
- Slide 20
- And all of the points in this square
- Slide 21
- are transformed to all of the points in this parallelogram
- Slide 22
- |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
- Slide 23
- Once again, knowing where Mr. Matrix sends Eee-Juan and Eee-too
actually tells us everything.
- Slide 24
- And this even applies to me
- Slide 25
- First realize that, amusing as I am, Im actually just some
points in the plane: line segments and circles.
- Slide 26
- So, all of my points move under the instructions of Mr.
Matrix.
- Slide 27
- Every one of my points is just a sum of some amount of Eee-Juan
and some amount of Eee-too.
- Slide 28
- This is where Mr. Matrix sends my points.
- Slide 29
- We are going to see what happens to me with various versions of
Mr. Matrix.
- Slide 30
- You should pay attention to what happens to my line segments
and circles and this box around me.
- Slide 31
- But before that, notice that I am not symmetric: one arm is
raised the other arm isnt.
- Slide 32
- Pay special attention to the two arms.
- Slide 33
- So here we go. First, Mr. Matrix is the identity matrix. Mr.
Matrix as the identity
- Slide 34
- and he transforms me to
- Slide 35
- Slide 36
- Yup. No change whatsoever.
- Slide 37
- Pretty boring. Right? Written as I
- Slide 38
- This time Mr. Matrix is just half of what he was as the
identity matrix. Written as I
- Slide 39
- and he transforms me to
- Slide 40
- Slide 41
- (back to blue)
- Slide 42
- Ive been shrunk in half.
- Slide 43
- This is called a scaling. Notice the constant on the diagonal
of Mr. Matrix.
- Slide 44
- Lets change that constant to 2. Written as 2 I
- Slide 45
- Slide 46
- And now I am back to my original self. Notice the second
process undid what the first did.
- Slide 47
- The two processes are inverses of each other. ( I) -1 = 2
I
- Slide 48
- and if we were to apply this scaling again to me
- Slide 49
- Slide 50
- Slide 51
- ... I get twice as big. Same shape just twice as big.
- Slide 52
- Now lets see what this one does with one 2 and one 1.
- Slide 53
- Can you see Ive been stretched?
- Slide 54
- My x-component s have been doubled but my y-components were
left alone.
- Slide 55
- My head is no longer a circle but an ellipse.
- Slide 56
- The box around me is still a rectangle just twice as wide.
- Slide 57
- Im back to regular and now well reverse the positions of the 1
and 2.
- Slide 58
- Slide 59
- Slide 60
- My y-component s have been doubled but x- components were left
alone.
- Slide 61
- Again my head is an ellipse.
- Slide 62
- and again the box around me is still a rectangle now twice as
tall.
- Slide 63
- Back to normal. Now lets double the x-coordinate and halve the
y-coordinate at the same time. Notice the 2 and the .
- Slide 64
- Big time squishing, right? The box is twice as wide and half as
tall so the area is the same as before.
- Slide 65
- Lets go the other way: halve the x-coordinate and double the y-
coordinate. The 2 and the are switched.
- Slide 66
- Slide 67
- Those transformations stretched or shrank the x- or
y-coordinate or both.
- Slide 68
- Mr. Matrix was diagonal: non-zeros only in the upper left and
lower right positions.
- Slide 69
- Now lets go back to the identity - but add a non- zero in the
upper right. The upper right is now 1/2.
- Slide 70
- Slide 71
- The y-coordinates are left alone. The x- plus one half the
y-coordinates are added to get the new x-coordinates.
- Slide 72
- This is called a shear.
- Slide 73
- 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.
- Slide 74
- Slide 75
- The x-coordinates are left alone. The y- plus one half the
x-coordinates are added to get the new y-coordinates.
- Slide 76
- Moving on So what will this do? It looks sort of like the
identity. The 1s and 0s are reversed from the identity
- Slide 77
- Slide 78
- Do you believe Ive been rotated?
- Slide 79
- Look closer. Look at the arm I have raised. Is this really a
rotation?
- Slide 80
- Nope. Its a reflection. My x- and y- components have been
reversed.
- Slide 81
- This is easier to see if I draw in this 45 degree line.
- Slide 82
- A reflection is a flipping across some line. I am a mirror
image of my former self.
- Slide 83
- But other than that exactly the same: no shrinking, no
stretching.
- Slide 84
- 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?
- Slide 85
- This is a rotation through 90 degrees.
- Slide 86
- Notice it is not a reflection - not a mirror image.
- Slide 87
- Quiz Time: Watch this - is it a reflection or a rotation? Two
-1s
- Slide 88
- Slide 89
- This is a reflection. Do you see that it is a mirror image
across the line?
- Slide 90
- 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
- Slide 91
- So what is a general rotation?
- Slide 92
- This matrix performs a counterclockwise rotation of an angle
The last example had = /2 or 90 degrees
- Slide 93
- Moving counterclockwise is considered the positive
direction.
- Slide 94
- Lets try this rotation for = /10 or 18 degrees.
- Slide 95
- Slide 96
- and again
- Slide 97
- Slide 98
- Slide 99
- Slide 100
- Slide 101
- Slide 102
- You get the idea. If we call this matrix R, then the total
effect is R 7.
- Slide 103
- Slide 104
- Finally, we will see what happens when Mr. Matrix transforms me
over and over.
- Slide 105
- This is a special matrix called a stochastic matrix: no
negative numbers and each column has a sum of 1. Stochastic
Matrix
- Slide 106
- It is sometimes used to describe the probabilities of movements
between states.
- Slide 107
- Heres a state diagram corresponding to this matrix A B 4%
84%96% 16%
- Slide 108
- 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%
- Slide 109
- Applying Mr. Matrix over and over is a way of finding the
steady state. A B 4% 84%96% 16%
- Slide 110
- But lets see what happens when Mr. Matrix is applied over and
over to me.
- Slide 111
- Slide 112
- Slide 113
- Slide 114
- Slide 115
- Slide 116
- Slide 117
- Slide 118
- Slide 119
- Slide 120
- Slide 121
- Slide 122
- Slide 123
- And lets skip forward an infinite number of steps to
- Slide 124
- And now Im fixed. All of my points are called eigenvectors
corresponding to eigenvalue 1.
- Slide 125
- Timmy Twospace signing off. Bye.