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Math 5270
Transformational Geometry
Day 9
Summer 13
Day 9, Summer 13
Math 5270 Transformational Geometry
1/24
Composition of rotations
T1
T2
x
y
x
y
cos θ1 − sin θ1
sin θ1 cos θ2
cos θ2
sin θ2
=
=
x
·
y
− sin θ2
x
·
cos θ2
y
We said that one way of thinking of T1 ◦ T2 gives us the matrix:
Day 9, Summer 13
cos (θ1 + θ2 ) − sin (θ1 + θ2 )
sin (θ1 + θ2 ) cos (θ1 + θ2 )
x
·
y
Math 5270 Transformational Geometry
3/24
What does T1
Day 9, Summer 13
T2
y
give
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4/24
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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
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Math 5270 Transformational Geometry
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Matrix
of awith
linear
transformation
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Math 5270 Transformational Geometry
6/24
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For
each ofwith
thea following
matrices,
describe the
effects of
its transformation
(a)
3 0
0 3
"5
−1 0
13
(b)
(c) 12
0 −1
13
12
− 13
5
12
#
3 −4
−1 16
(d)
(e)
4 3
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?
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Math 5270 Transformational Geometry
7/24
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Perform
the
following
multiplication
using the- www.PDFAnno
definition
given yesterday then interpret the process using
transformations
Day 9, Summer 13
 
1
1 2 3  
2
3 2 1
3
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Find
the entries
of the
following
matrices:
1
2
the 2 × 2 matrix M for the reflection across the line y = x.
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?
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Investigating
a matrix
The matrix M =
" 3
−5
4
5
4
5
3
5
#
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.
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Math 5270 Transformational Geometry
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Commuting
matrices
−0.6 0.8
0.8 0.6
Matrices M =
and N =
represent
0.8 0.6
0.6 −0.8
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?
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Can
you find
a matrix
for theofreflection
in y =- www.PDFAnno
ax?
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Math 5270 Transformational Geometry
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