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

]


What does T1

(T

2

([y

]))give


com

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


Matrix of a linear transformation



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?



Perform the following multiplication using the definitiongiven yesterday then interpret the process usingtransformations

[1 2 3

3 2 1

]1

2

3



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?



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.



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?



Can you find a matrix for the reflection in y = ax?



Math 5270 Transformational Geometryemina/teaching/5270s13/5270_Day9_notes.pdf · Day 9, Summer 13...

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