Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS Lesson 9: Composition of Linear Transformations Student Outcomes ο§ Students use technology to perform compositions of linear transformations in β3 . Lesson Notes In Lesson 8, students discovered that if they compose two linear transformations in the plane, πΏπ΄ and πΏπ΅ , represented by matrices π΄ and π΅, respectively, then the resulting transformation can be produced using a single matrix π΄π΅. In this lesson, this result from transformations in the plane is extended to transformations in three-dimensional space, β3 . This lesson was designed to be implemented using the GeoGebra applet TransformCubes (http://eureka-math.org/G12M2L9/geogebra-TransformCubes), which allows students to visualize the transformation of the cube. Classwork Opening Exercise (10 minutes) The Opening Exercise reviews the discovery from the Problem Set in Lesson 7 that the matrix of a transformation in β3 is determined by the images of the three points (1,0,0), (0,1,0), and (0,0,1). This fact is needed to construct matrices of transformations in this lesson before composing them. Students may question what is meant by counterclockwise rotation in space; if this happens, let them know that a rotation about the π§-axis is considered to be a counterclockwise rotation if it rotates the π₯π¦-plane counterclockwise when placed in its standard orientation as shown to the right. Students should work these exercises with pencil and paper. Opening Exercise Recall from Problem 1, part (d), of the Problem Set of Lesson 7 that if you know what a linear transformation does to the three points (π, π, π), (π, π, π), and (π, π, π), you can find the matrix of the transformation. How do the images of these three points lead to the matrix of the transformation? a. Suppose that a linear transformation π³π rotates the unit cube by ππ° counterclockwise about the π-axis. Find the matrix π¨π of the transformation π³π . Since this transformation rotates by ππ° counterclockwise in the ππ-plane, a vector along the positive π-axis will be transformed to lie along the positive π-axis, a vector along the positive π-axis will be transformed to lie along the negative π-axis, and a vector along the π-axis will be left alone. Thus, Scaffolding: Encourage struggling students to draw the image of a cube before and after the transformation to find the images of the points (1,0,0), (0,1,0), and (0,0,1) in space. π π π −π π π π³π ([π]) = [π], π³π ([π]) = [ π ], and π³π ([π]) = [π], π π π π π π π −π π so the matrix of the transformation is π¨π = [π π π]. π π π Lesson 9: 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 150 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS b. Suppose that a linear transformation π³π rotates the unit cube by ππ° counterclockwise about the π-axis. Find the matrix π¨π of the transformation π³π . Since this transformation rotates by ππ° counterclockwise in the ππ-plane, a vector along the positive π-axis will be transformed to lie along the positive π-axis, a vector along the π-axis will be left alone, and a vector along the positive π-axis will be transformed to lie along the negative π-axis. Thus, π π π π π −π π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [ π ], π π π π π π π π −π so the matrix of the transformation is π¨π = [π π π ]. π π π c. Suppose that a linear transformation π³π scales by π in the π-direction, scales by π in the π-direction, and scales by π in the π-direction. Find the matrix π¨π of the transformation π³π . Since this transformation scales by π in the π-direction, by π in the π-direction, and by π in the π-direction, a vector along the π-axis will be multiplied by π, a vector along the π-axis will be multiplied by π, and a vector along the π-axis will be multiplied by π. Thus, π π π π π π π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [π], π π π π π π π π π so the matrix of the transformation is π¨π = [π π π]. π π π d. Suppose that a linear transformation π³π projects onto the ππ-plane. Find the matrix π¨π of the transformation π³π . Since this transformation projects onto the ππ-plane, a vector along the π-axis will be left alone, a vector along the π-axis will be left alone, and a vector along the π-axis will be transformed into the zero vector. Thus, π π π π π π π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [π], π π π π π π π π π so the matrix of the transformation is π¨π = [π π π]. π π π e. Suppose that a linear transformation π³π projects onto the ππ-plane. Find the matrix π¨π of the transformation π³π . Since this transformation projects onto the ππ-plane, a vector along the π-axis will be left alone, a vector along the π-axis will be transformed into the zero vector, and a vector along the π-axis will be left alone. Thus, π π π π π π π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [π], π π π π π π π π π so the matrix of the transformation is π¨π = [π π π]. π π π Lesson 9: 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 151 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS f. Suppose that a linear transformation π³π reflects across the plane with equation π = π. Find the matrix π¨π of the transformation π³π . Since this transformation reflects across the plane π = π, a vector along the positive π-axis will be transformed into a vector along the positive π-axis with the same length, a vector along the positive π-axis will be transformed into a vector along the positive π-axis, and a vector along the π-axis will be left alone. Thus, π π π π π π π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [π], π π π π π π π π π so the matrix of the transformation is π¨π = [π π π]. π π π g. π π π π π π Suppose that a linear transformation π³π satisfies π³π ([π]) = [π], π³π ([π]) = [π], and π³π ([π]) = [π ]. Find π π π π π π π the matrix π¨π of the transformation π³π . What is the geometric effect of this transformation? π π π The matrix of this transformation is π¨π = [π π π]. The geometric effect of π³π is to stretch by a factor of π π π π in the π-direction and scale by a factor of h. π π π in the π-direction. π π π −π π π Suppose that a linear transformation π³π satisfies π³π ([π]) = [π] , π³π ([π]) = [ π ], and π³π ([π]) = [π]. π π π π π π Find the matrix of the transformation π³π . What is the geometric effect of this transformation? π −π π The matrix of this transformation is π¨π = [π π π]. The geometric effect of π³π is to rotate by ππ° in the π π π ππ-plane, to scale by √π in both the π and π directions, and to not change in the π-direction. Once students have completed the Opening Exercise with pencil and paper, allow them to use the GeoGebra applet TransformCubes.ggb to check their work before continuing. The remaining exercises rely on the matrices π΄1 –π΄8 , so ensure that students have found the correct matrices before proceeding with the lesson. Discussion (2 minutes) ο§ In Lesson 8, we saw that for linear transformations in the plane, if πΏπ΄ is a linear transformation represented by a 2 × 2 matrix π΄, and πΏπ΅ is a linear transformation represented by a 2 × 2 matrix π΅, then the 2 × 2 matrix π΄π΅ is the matrix of the composition of πΏπ΅ followed by πΏπ΄ . Today we explore composition of linear transformations in β3 to see whether the same result extends to the case when π΄, π΅, and π΄π΅ are 3 × 3 matrices. Lesson 9: 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 152 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS Exploratory Challenge 1 (25 minutes) In the Exploratory Challenge, students predict the geometric effect of composing pairs of transformations from the Opening Exercise and then check their predictions with the GeoGebra applet TransformCubes.ggb. Exploratory Challenge 1 Transformations π³π –π³π refer to the linear transformations from the Opening Exercise. For each pair: MP.3 a. i. Make a conjecture to predict the geometric effect of performing the two transformations in the order specified. ii. Find the product of the corresponding matrices in the order that corresponds to the indicated order of composition. Remember that if we perform a transformation π³π© with matrix π© and then π³π¨ with matrix π¨, the matrix that corresponds to the composition π³π¨ β π³π© is π¨π©. That is, π³π© is applied first, but matrix π© is written second. iii. Use the GeoGebra applet TransformCubes.ggb to draw the image of the unit cube under the transformation induced by the matrix product in part (ii). Was your conjecture in part (i) correct? Perform π³π and then π³π . i. ii. iii. b. Since π³π reflects across the plane through π = π that is perpendicular to the ππ-plane, performing π³π twice in succession will result in the identity transformation. π π π π π π π π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π]. Since π¨π ⋅ π¨π is the identity matrix, we know that π π π π π π π π π π³π β π³π is the identity transformation. The conjecture was correct. Perform π³π and then π³π . i. ii. iii. Sample student response: Since π³π rotates ππ° about the π-axis and π³π rotates ππ° about the π-axis, the composition π³π β π³π should rotate πππ° about the line π = −π in the ππ-plane. π π −π π −π π π π −π π¨π ⋅ π¨π = [π π π ] ⋅ [π π π] = [π π π] π π π π π π π −π π The conjecture in part (i) is not correct. While it appears that the composition π³π β π³π is a rotation by πππ° about the line π = −π, it is not because, for example, point (π, π, π) is transformed to point (π, π, −π) and does not remain on the π-axis after the transformation. Thus, this cannot be a rotation around the π-axis. Lesson 9: 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 153 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS c. Perform π³π and then π³π . i. ii. iii. d. π π π π π π π π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π The conjecture in part (i) is correct. The cube is first transformed to a square in the ππ-plane and then transformed onto a segment on the π-axis. Perform π³π and then π³π . i. ii. iii. e. Since π³π projects onto the ππ-plane and π³π projects onto the ππ-plane, the composition π³π β π³π will project onto the π-axis. Since π³π projects onto the ππ-plane and π³π scales in the π, π, and π directions, the composition π³π β π³π will project onto the ππ-plane and scale in the π and π directions. π π π π π π π π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π The conjecture in part (i) is correct. Perform π³π and then π³π . i. Since π³π scales in the π, π, and π directions and so does π³π , the composition will be a transformation that also scales in all three directions but with different scale factors. ii. π π π π π π π π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π π iii. The conjecture from (i) is correct. Lesson 9: 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 154 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS f. Perform π³π and then π³π . i. ii. iii. g. π π π π −π π π −π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π The conjecture from part (i) is correct. Perform π³π and then π³π . i. ii. iii. h. Transformation π³π rotates the unit cube by ππ° about the π-axis and stretches by a factor of √π in both the π and π directions, while π³π projects the image onto the ππ-plane. The composition will transform the unit cube into a larger square that has been rotated ππ° in the ππ-plane. Transformation π³π projects onto the ππ-plane, and transformation π³π reflects across the plane through the line π = π in the ππ-plane and is perpendicular to the ππ-plane, so the composition π³π β π³π will appear to be the reflection in the ππ-plane across the line π = π. π π π π π π π π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π The conjecture in part (i) is correct. Perform π³π and then π³π . i. Since π³π rotates ππ° around the π-axis and π³π scales in the π and π directions, the composition π³π β π³π will rotate and scale simultaneously. ii. π π π π π −π π π −π π¨π ⋅ π¨π = [π π π] ⋅ [π π π ] = [π π π ] π π π π π π π π π π π iii. The conjecture in part (i) is correct. Lesson 9: 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 155 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS i. Perform π³π and then π³π . i. ii. iii. ο§ Since π³π rotates by ππ° about the π-axis and scales by √π in the π and π directions, performing this transformation twice will rotate by ππ° about the π-axis and scale by π in the π and π directions. π −π π π −π π π −π π π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π] π π π π π π π π π The conjecture in part (i) is correct. Do a 30-second Quick Write on what we have discovered in Exploratory Challenge 1, and share with your neighbor. Exploratory Challenge 2 (Optional) This optional challenge is for students who finished Exploratory Challenge 1 early. The challenge is designed to prompt the question of whether or not order matters when composing two linear transformations, a question that is definitively answered in the next lesson and demonstrates that matrix multiplication is in general not commutative. Students are asked to compose two linear transformations πΏπ΄ and πΏπ΅ , with matrices π΄ and π΅, respectively, and to compare πΏπ΄ β πΏπ΅ with πΏπ΅ β πΏπ΄ . The directions for this challenge are left intentionally vague so that students may use either an algebraic or a geometric approach to answer the question. Exploratory Challenge 2 Transformations π³π –π³π refer to the transformations from the Opening Exercise. For each of the following pairs of matrices π¨ and π© below, compare the transformations π³π¨ β π³π© and π³π© β π³π¨ . a. π³π and π³π π π π π π π π π π Transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π], and π π π π π π π π π π π π π π π π π π transformation π³π β π³π has matrix representation π¨π β π¨π = [π π π] ⋅ [π π π] = [π π π]. π π π π π π π π π Since the two transformations have the same matrix representation, they are the same transformation: π³π β π³π = π³π β π³π Lesson 9: 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 156 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS b. π³π and π³π π π −π π π π π π −π Transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π ] ⋅ [π π π] = [π π π ], and π π π π π π π π π π π π π π −π π π −π transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π ] = [π π π ]. π π π π π π π π π Since the two transformations have the same matrix representation, they are the same transformation: π³π β π³π = π³π β π³π c. π³π and π³π π π π π π π π π π Transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π], and π π π π π π π π π π π π π π π π π π π transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π]. π π π π π π π π π π Since the two transformations have the same matrix representation, they are the same transformation: π³π β π³π = π³π β π³π d. π³π and π³π π π π π π π π π π Transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π], and π π π π π π π π π π π π π π π π π π transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π]. π π π π π π π π π Since the two transformations have different matrix representations, they are not the same transformation: π³π β π³π ≠ π³π β π³π e. π³π and π³π π π π π −π π π −π π Transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π], and π π π π π π π π π π π π π π π −π π π −π π transformation π³π β π³π has matrix representation π¨π ⋅ π¨π = [π π π] ⋅ [π π π] = [π π π]. π π π π π π π π π π π Since the two transformations have different matrix representations, they are not the same transformation: π³π β π³π ≠ π³π β π³π f. What can you conclude about the order in which you compose two linear transformations? In some cases, the order of composition of two linear transformations matters. For two matrices π¨ and π©, the transformation π³π¨ β π³π© is not always the same transformation as π³π© β π³π¨ . Lesson 9: 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 157 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS Closing (4 minutes) Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements are listed below. Lesson Summary ο§ The linear transformation induced by a π × π matrix π¨π© has the same geometric effect as the sequence of the linear transformation induced by the π × π matrix π© followed by the linear transformation induced by the π × π matrix π¨. ο§ That is, if matrices π¨ and π© induce linear transformations π³π¨ and π³π© in βπ , respectively, then the linear π π transformation π³π¨π© induced by the matrix π¨π© satisfies π³π¨π© ([π]) = π³π¨ (π³π© ([π])). π π Exit Ticket (4 minutes) Lesson 9: 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 158 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS Name Date Lesson 9: Composition of Linear Transformations Exit Ticket Let π΄ be the matrix representing a rotation about the π§-axis of 45° and π΅ be the matrix representing a dilation of 2. a. Write down π΄ and π΅. b. 3 Let π₯ = [−1]. Find the matrix representing a dilation of π₯ by 2 followed by a rotation about the π§-axis of 45°. 2 c. Do your best to sketch a picture of π₯, π₯ after the first transformation, and π₯ after both transformations. You may use technology to help you. Lesson 9: 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 159 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS Exit Ticket Sample Solutions Let π¨ be the matrix representing a rotation about the π-axis of ππ° and π© be the matrix representing a dilation of π. a. Write down π¨ and π©. √π √π − π π π π π π π¨ = √π √π , π© = [π π π] π π π π π π [π π π] b. π Let π = [−π]. Find the matrix representing a dilation of π by π followed by a rotation about the π-axis of π ππ°. √π −√π π π¨π© = [√π √π π] π π π c. Do your best to sketch a picture of π, π after the first transformation, and π after both transformations. You may use technology to help you. Lesson 9: 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 160 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS Problem Set Sample Solutions 1. π Let π¨ be the matrix representing a dilation of , and let π© be the matrix representing a reflection across the π ππ-plane. a. Write π¨ and π©. π π π π −π π π π π¨= π π , π© = [ π π π] π π π π π π π [ π] b. Evaluate π¨π©. What does this matrix represent? π π π π π π¨π© = π π π π [ π π π] − π π¨π© is a reflection across the ππ-plane followed by a dilation of . π c. −π π π Let π = [π], π = [ π ], and π = [−π]. Find (π¨π©)π, (π¨π©)π, and (π¨π©)π. π π −π π π −π − π (π¨π©)π = [ π] , (π¨π©)π = π , (π¨π©)π = [−π] π −π π π [π] 2. Let π¨ be the matrix representing a rotation of ππ° about the π-axis, and let π© be the matrix representing a dilation of π. a. Write π¨ and π©. π π π π √π π π π π − π¨= π π , π© = [π π π] π π π π √π [π π π ] b. Evaluate π¨π©. What does this matrix represent? π π π π√π π π − π¨π© = π π π π√π [π π π ] π¨π© is a dilation of π followed by a rotation of ππ° about the π-axis. Lesson 9: 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 161 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS c. π π π Let π = [π], π = [π], π = [π]. Find (π¨π©)π, (π¨π©)π, and (π¨π©)π. π π π π (π¨π©)π = [π] π π π√π (π¨π©)π = π π [ π ] π π π (π¨π©)π = π√π [ π ] − 3. Let π¨ be the matrix representing a dilation of π, and let π© be the matrix representing a reflection across the plane π = π. a. Write π¨ and π©. π π π π π π π¨ = [π π π] , π© = [π π π] π π π π π π b. Evaluate π¨π©. What does this matrix represent? π π π π¨π© = [π π π] π π π π¨π© is a reflection across the π = π plane followed by a dilation of π. c. −π Let π = [ π ]. Find (π¨π©)π. π ππ (π¨π©)π = [−π] π 4. π π π π −π π Let π¨ = [π π π], π© = [π π π]. π π π π π π a. Evaluate π¨π©. π −π π [π −π π] π π π b. −π Let π = [ π ]. Find (π¨π©)π. π −π [−ππ] π Lesson 9: 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 162 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM M2 PRECALCULUS AND ADVANCED TOPICS c. Graph π and (π¨π©)π. (π΄π΅)π₯ π₯ 5. π π π π π Let π¨ = π π π , π© = [π π π [π π π] a. π π −π π]. π π Evaluate π¨π©. π π π π −π π π [π π π] π b. π Let π = [π]. Find (π¨π©)π. π π −π [ ππ ] π Lesson 9: 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 163 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS c. 6. Graph π and (π¨π©)π. π π π π π π Let π¨ = [π π π], π© = [π π π]. π π π π π π a. Evaluate π¨π©. π π π [π π π] π π π b. π Let π = [−π]. Find (π¨π©)π. π π [−π] π Lesson 9: 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 164 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M2 PRECALCULUS AND ADVANCED TOPICS c. Graph π and (π¨π©)π. d. What does π¨π© represent geometrically? π¨π© represents a projection onto the ππ-plane, and a dilation of π in the π-direction and π in the π-direction. 7. Let π¨, π©, πͺ be π × π matrices representing linear transformations. a. What does π¨(π©πͺ) represent? The linear transformation of applying the linear transformation that πͺ represents followed by the transformation that π© represents followed by the transformation that π¨ represents b. MP.7 Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left? Yes, in general. When you multiply by a matrix on the left, you are applying a linear transformation after all linear transformations to the right have been applied. c. Does (π¨π©)πͺ represent something different from π¨(π©πͺ)? Explain. No, it does not. This is the linear transformation obtained by applying C and then π¨π©, which is π© followed by π¨. 8. π Let π¨π© represent any composition of linear transformations in βπ . What is the value of (π¨π©)π where π = [π]? π Since a composition of linear transformations in βπ is also a linear transformation, we know that applying it to the origin will result in no change. Lesson 9: 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 165 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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