Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
Classwork
Opening Exercise
Compute the product 𝐴𝐵 for the following pairs of matrices.
a.
0
𝐴=[
1
−1
0
], 𝐵 = [
0
1
b.
1
𝐴=[
0
0
−1
], 𝐵 = [
−1
0
√2
c.
d.
e.
0
]
1
√2
− 2
4 0
], 𝐵 = [
]
𝐴=[2
√2
2
√2
0
4
2
1
𝐴=[
0
1
2
], 𝐵 = [
1
0
1
𝐴=[ 2
−
−1
]
0
√3
2
√3
0
]
2
√3
2 ], 𝐵 = [ 2
1
1
2
2
Lesson 8:
1
2]
√3
2
−
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.56
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Exploratory Challenge
a.
For each pair of matrices 𝐴 and 𝐵 given below:
1.
𝑥
𝑥
Describe the geometric effect of the transformation 𝐿𝐵 ([𝑦]) = 𝐵 ⋅ [𝑦].
2.
𝑥
𝑥
Describe the geometric effect of the transformation 𝐿𝐴 ([𝑦]) = 𝐴 ⋅ [𝑦].
3.
𝑥
𝑥
Draw the image of the unit square under the transformation 𝐿𝐵 ([𝑦]) = 𝐵 ⋅ [𝑦].
4.
𝑥
𝑥
Draw the image of the transformed square under the transformation 𝐿𝐴 ([𝑦]) = 𝐴 ⋅ [𝑦].
5.
Describe the geometric effect on the unit square of performing first 𝐿𝐵 and then 𝐿𝐴 .
6.
Compute the matrix product 𝐴𝐵.
7.
𝑥
𝑥
Describe the geometric effect of the transformation 𝐿𝐴𝐵 ([𝑦]) = 𝐴𝐵 ⋅ [𝑦 ].
i.
𝐴=[
0
1
−1
0
], 𝐵 = [
0
1
Lesson 8:
−1
]
0
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.57
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M2
PRECALCULUS AND ADVANCED TOPICS
ii.
𝐴=[
1
0
0
−1
], 𝐵 = [
−1
0
√2
iii.
𝐴=[2
√2
2
−
0
]
1
√2
2 ], 𝐵 = [4
√2
0
0
]
4
2
Lesson 8:
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.58
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M2
PRECALCULUS AND ADVANCED TOPICS
1
0
1
2
], 𝐵 = [
1
0
iv.
𝐴=[
v.
1
𝐴=[ 2
−
√3
2
√3
0
]
2
√3
2 ], 𝐵 = [ 2
1
1
2
2
Lesson 8:
1
2]
√3
2
−
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.59
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M2
PRECALCULUS AND ADVANCED TOPICS
b.
Make a conjecture about the geometric effect of the linear transformation produced by the matrix 𝐴𝐵. Justify
your answer.
Lesson 8:
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.60
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M2
PRECALCULUS AND ADVANCED TOPICS
Lesson Summary
The linear transformation produced by matrix 𝐴𝐵 has the same geometric effect as the sequence of the linear
transformation produced by matrix 𝐵 followed by the linear transformation produced by matrix 𝐴.
That is, if matrices 𝐴 and 𝐵 produce linear transformations 𝐿𝐴 and 𝐿𝐵 in the plane, respectively, then the linear
transformation 𝐿𝐴𝐵 produced by the matrix 𝐴𝐵 satisfies
𝑥
𝑥
𝐿𝐴𝐵 ([𝑦]) = 𝐿𝐴 (𝐿𝐵 ([𝑦])).
Problem Set
1.
1
Let 𝐴 be the matrix representing a dilation of , and let 𝐵 be the matrix representing a reflection across the 𝑦-axis.
2
a.
Write 𝐴 and 𝐵.
b.
Evaluate 𝐴𝐵. What does this matrix represent?
−1
8
5
Let 𝑥 = [ ], 𝑦 = [ ], and 𝑧 = [ ]. Find (𝐴𝐵)𝑥, (𝐴𝐵)𝑦, and (𝐴𝐵)𝑧.
3
−2
6
c.
2.
Let 𝐴 be the matrix representing a rotation of 30°, and let 𝐵 be the matrix representing a dilation of 5.
a.
Write 𝐴 and 𝐵.
b.
Evaluate 𝐴𝐵. What does this matrix represent?
1
Let 𝑥 = [ ]. Find (𝐴𝐵)𝑥.
0
c.
3.
Let 𝐴 be the matrix representing a dilation of 3, and let 𝐵 be the matrix representing a reflection across the line
𝑦 = 𝑥.
a.
Write 𝐴 and 𝐵.
b.
Evaluate 𝐴𝐵. What does this matrix represent?
−2
Let 𝑥 = [ ]. Find (𝐴𝐵)𝑥.
7
c.
4.
3 0
0 −1
] and 𝐵 = [
].
3 3
1 0
Evaluate 𝐴𝐵.
−2
Let 𝑥 = [ ]. Find (𝐴𝐵)𝑥.
2
Graph 𝑥 and (𝐴𝐵)𝑥.
Let 𝐴 = [
a.
b.
c.
Lesson 8:
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.61
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M2
PRECALCULUS AND ADVANCED TOPICS
5.
Let 𝐴 = [
a.
b.
c.
6.
b.
c.
8.
0
2
1
3
3
] and 𝐵 = [
1
1
].
−3
Evaluate 𝐴𝐵.
0
Let 𝑥 = [ ]. Find (𝐴𝐵)𝑥.
3
Graph 𝑥 and (𝐴𝐵)𝑥.
2 2
0 2
] and 𝐵 = [
].
2 0
2 2
Evaluate 𝐴𝐵.
3
Let 𝑥 = [ ]. Find (𝐴𝐵)𝑥.
−2
Graph 𝑥 and (𝐴𝐵)𝑥.
Let 𝐴 = [
a.
7.
1
3
Let 𝐴, 𝐵, 𝐶 be 2 × 2 matrices representing linear transformations.
a.
What does 𝐴(𝐵𝐶) represent?
b.
Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left?
c.
Does (𝐴𝐵)𝐶 represent something different from 𝐴(𝐵𝐶)? Explain.
0
Let 𝐴𝐵 represent any composition of linear transformations in ℝ2 . What is the value of (𝐴𝐵)𝑥 where 𝑥 = [ ]?
0
Lesson 8:
Composition of Linear Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M2-TE-1.3.0-08.2015
S.62
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.