Math 5270 Transformational Geometryemina/teaching/5270s13/5270_Day9_notes.pdf · Day 9, Summer 13...
Transcript of Math 5270 Transformational Geometryemina/teaching/5270s13/5270_Day9_notes.pdf · Day 9, Summer 13...
Math 5270Transformational Geometry
Day 9
Summer 13
Day 9, Summer 13 Math 5270 Transformational Geometry 1/24
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Composition of rotations
T1
([x
y
])=
[cos θ1 − sin θ1sin θ1 cos θ2
]·[
x
y
]
T2
([x
y
])=
[cos θ2 − sin θ2sin θ2 cos θ2
]·[
x
y
]We said that one way of thinking of T1 ◦ T2 gives us the matrix:[
cos (θ1 + θ2) − sin (θ1 + θ2)
sin (θ1 + θ2) cos (θ1 + θ2)
]·[
x
y
]
Day 9, Summer 13 Math 5270 Transformational Geometry 3/24
What does T1
(T
2
([y
]))give
Day 9, Summer 13 Math 5270 Transformational Geometry 4/24
com
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Rearrange yourselves into the following groups:
Kaitlyn, Vivian, Chris M,
Jessie, Diane, James, Lisa
Lisett, Anthony, Johnny, Tyler
Erika, Neiko, Annie, Becky
Shinil, Rick, Michele,
Mary Ch, Anna, Kassie
Kelli, Mary C, Mike, Stacey
Elsina, Chris S, Paul, Daniel
Day 9, Summer 13 Math 5270 Transformational Geometry 5/24
Matrix of a linear transformation
Day 9, Summer 13 Math 5270 Transformational Geometry 6/24
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For each of the following matrices, describe the effects ofits transformation
(a)
[3 0
0 3
](b)
[−1 0
0 −1
](c)
[ 513 −12
131213
512
](d)
[3 −4
4 3
](e)
[−1 1
6
2 −13
]
Related question: Which one of the preceding matrices represent
isometries? In order for a matrix to represent isometry, what must
be true of its column vectors? Why does that guarantee an
isometry?
Day 9, Summer 13 Math 5270 Transformational Geometry 7/24
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Perform the following multiplication using the definitiongiven yesterday then interpret the process usingtransformations
[1 2 3
3 2 1
]1
2
3
Day 9, Summer 13 Math 5270 Transformational Geometry 8/24
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Find the entries of the following matrices:
1 the 2× 2 matrix M for the reflection across the line y = x .
2 the 2× 2 matrix N for the 90◦ counterclockwise rotation
about the origin.
3 the product MN; what transformation does this represent?
4 the product NM; what transformation does this represent?
5 the product MM; what transformation does this represent?
Day 9, Summer 13 Math 5270 Transformational Geometry 9/24
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Investigating a matrix
The matrix M =
[−3
545
45
35
]defines an isometry of the xy - plane.
1 What special properties do the column vectors of this matrix
have?
2 Verify that the point (2, 4) remains stationary when M is
applied to it. What might M be?
3 What is MM? What does this suggest about the geometric
transformation that M represents.
Day 9, Summer 13 Math 5270 Transformational Geometry 10/24
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Commuting matrices
Matrices M =
[−0.6 0.8
0.8 0.6
]and N =
[0.8 0.6
0.6 −0.8
]represent
reflections in the lines y = 2x and 3y = x . Verify that MN is not
equal to NM, and explain why this should have been expected.
What transformations do the two products represent?
Day 9, Summer 13 Math 5270 Transformational Geometry 11/24
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Can you find a matrix for the reflection in y = ax?
Day 9, Summer 13 Math 5270 Transformational Geometry 12/24
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