Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Lesson 6: Linear Transformations as Matrices
Classwork
Opening Exercise
Let 𝐴 = (
7
5
𝑦1
𝑥1
−2
), 𝑥 = (𝑥 ), and 𝑦 = (𝑦 ). Does this represent a linear transformation? Explain how you know.
−3
2
2
Exploratory Challenge 1: The Geometry of 3-D Matrix Transformations
a.
What matrix in ℝ2 serves the role of 1 in the real number system? What is that role?
b.
What matrix in ℝ2 serves the role of 0 in the real number system? What is that role?
Lesson 6:
Linear Transformations as Matrices
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.40
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
c.
What is the result of scalar multiplication in ℝ2 ?
d.
Given a complex number 𝑎 + 𝑏𝑖, what represents the transformation of that point across the real axis?
Lesson 6:
Linear Transformations as Matrices
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.41
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
Exploratory Challenge 2: Properties of Vector Arithmetic
a.
Is vector addition commutative? That is, does 𝑥 + 𝑦 = 𝑦 + 𝑥 for each pair of points in ℝ2 ? What about points
in ℝ3 ?
b.
Is vector addition associative? That is, does (𝑥 + 𝑦) + 𝑟 = 𝑥 + (𝑦 + 𝑟) for any three points in ℝ2 ? What
about points in ℝ3 ?
Lesson 6:
Linear Transformations as Matrices
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.42
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
c.
Does the distributive property apply to vector arithmetic? That is, does 𝑘 ∙ (𝑥 + 𝑦) = 𝑘𝑥 + 𝑘𝑦 for each pair of
points in ℝ2 ? What about points in ℝ3 ?
d.
Is there an identity element for vector addition? That is, can you find a point 𝑎 in ℝ2 such that 𝑥 + 𝑎 = 𝑥 for
every point 𝑥 in ℝ2 ? What about for ℝ3 ?
Lesson 6:
Linear Transformations as Matrices
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.43
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
e.
Does each element in ℝ2 have an additive inverse? That is, if you take a point 𝑎 in ℝ2 , can you find a second
point 𝑏 such that 𝑎 + 𝑏 = 0?
Lesson 6:
Linear Transformations as Matrices
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.44
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Problem Set
1.
Show that the associative property, 𝑥 + (𝑦 + 𝑧) = (𝑥 + 𝑦) + 𝑧, holds for the following.
3
−4
−1
a. 𝑥 = ( ), 𝑦 = ( ), 𝑧 = ( )
−2
2
5
3
2
0
b. 𝑥 = (−2), 𝑦 = ( 5 ), 𝑧 = ( 0 )
1
−2
−3
2.
Show that the distributive property, 𝑘 (𝑥 + 𝑦) = 𝑘𝑥 + 𝑘𝑦, holds for the following.
−2
5
a. 𝑥 = ( ), 𝑦 = ( ), 𝑘 = −2
4
−3
b.
3.
4.
3
−4
𝑥 = (−2), 𝑦 = ( 6 ), 𝑘 = −3
5
−7
Compute the following.
1 0 2 2
a. (0 1 2) (1)
2 2 3 3
−1 2
3
1
b. ( 3
1 −2) (−2)
1 −2 3
1
1 0 2 2
c. (0 1 0) (1)
2 0 3 2
𝑎
3
Let 𝑥 = (1). Compute 𝐿(𝑥) = [𝑏
𝑐
2
−1 0 0
a. [ 0 1 0]
0 0 1
2 0 0
b. [0 2 0]
0 0 2
0 1 0
c. [1 0 0]
0 0 1
1 0 0
d. [0 0 1]
0 1 0
Lesson 6:
𝑑
𝑒
𝑓
𝑔
ℎ ] ∙ 𝑥, plot the points, and describe the geometric effect to 𝑥.
𝑖
Linear Transformations as Matrices
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.45
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
5.
6.
𝑎
3
Let 𝑥 = (1). Compute 𝐿(𝑥) = [𝑏
𝑐
2
0 0 0
a. [0 0 0]
0 0 0
3 0 0
b. [0 3 0]
0 0 3
−1 0
0
c. [ 0 −1 0 ]
0
0 −1
−1 0 0
d. [ 0 1 0]
0 0 1
1 0 0
e. [0 −1 0]
0 0 1
1 0 0
f. [0 1 0 ]
0 0 −1
0 1 0
g. [1 0 0]
0 0 1
1 0 0
h. [0 0 1]
0 1 0
0 0 1
i.
[0 1 0]
1 0 0
𝑔
ℎ ] ∙ 𝑥. Describe the geometric effect to 𝑥.
𝑖
1
Find the matrix that will transform the point 𝑥 = (3) to the following point:
2
−4
a. (−12)
−8
b.
7.
𝑑
𝑒
𝑓
3
( 1)
2
2
Find the matrix/matrices that will transform the point 𝐱 = (3) to the following point:
1
6
−1
a. 𝐱 ′ = (4)
b. 𝐱 ′ = ( 3 )
2
2
Lesson 6:
Linear Transformations as Matrices
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.46
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.