## Engage NY Eureka Math Precalculus Module 2 Lesson 9 Answer Key

### Eureka Math Precalculus Module 2 Lesson 9 Exercise Answer Key

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 (1,0,0), (0,1,0), and (0,0,1), 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 L_{1} rotates the unit cube by 90° counterclockwise about the z-axis. Find the matrix A_{1} of the transformation L_{1}.

Answer:

Since this transformation rotates by 90° counterclockwise in the xy-plane, a vector along the positive x-axis will be transformed to lie along the positive y-axis, a vector along the positive y-axis will be transformed to lie along the negative x-axis, and a vector along the z-axis will be left alone. Thus,

so the matrix of the transformation is A_{1}=\(\left[\begin{array}{ccc}

0 & -1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 1

\end{array}\right] .\)

b. Suppose that a linear transformation L_{2} rotates the unit cube by 90° counterclockwise about the y-axis. Find the matrix A_{2} of the transformation L_{2}.

Answer:

Since this transformation rotates by 90° counterclockwise in the xz-plane, a vector along the positive x-axis will be transformed to lie along the positive z-axis, a vector along the y-axis will be left alone, and a vector along the positive z-axis will be transformed to lie along the negative x-axis. Thus,

so the matrix of the transformation is A_{2}=\(\left[\begin{array}{ccc}

0 & 0 & -1 \\

0 & 1 & 0 \\

1 & 0 & 0

\end{array}\right]\).

c. Suppose that a linear transformation L_{3} scales by 2 in the x-direction, scales by 3 in the y-direction, and scales by 4 in the z-direction. Find the matrix A_{3} of the transformation L_{3}.

Answer:

Since this transformation scales by 2 in the x-direction, by 3 in the y-direction, and by 4 in the z-direction, a vector along the x-axis will be multiplied by 2, a vector along the y-axis will be multiplied by 3, and a vector along the z-axis will be multiplied by 4. Thus,

so the matrix of the transformation is A_{3}=\(\left[\begin{array}{lll}

2 & 0 & 0 \\

0 & 3 & 0 \\

0 & 0 & 4

\end{array}\right]\).

d. Suppose that a linear transformation L_{4} projects onto the xy-plane. Find the matrix A_{4} of the transformation L_{4}.

Answer:

Since this transformation projects onto the xy-plane, a vector along the x-axis will be left alone, a vector along the y-axis will be left alone, and a vector along the z-axis will be transformed into the zero vector. Thus,

so the matrix of the transformation is A_{4}=\(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 0

\end{array}\right]\).

e. Suppose that a linear transformation L_{5} projects onto the xz-plane. Find the matrix A_{5} of the transformation L_{5}.

Answer:

Since this transformation projects onto the xz-plane, a vector along the x-axis will be left alone, a vector along the y-axis will be transformed into the zero vector, and a vector along the z-axis will be left alone. Thus,

so the matrix of the transformation is A_{5}=\(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & 1

\end{array}\right]\).

f. Suppose that a linear transformation L_{6} reflects across the plane with equation y=x. Find the matrix A_{6} of the transformation L_{6}.

Answer:

Since this transformation reflects across the plane y=x, a vector along the positive x-axis will be transformed into a vector along the positive y-axis with the same length, a vector along the positive y-axis will be transformed into a vector along the positive x-axis, and a vector along the z-axis will be left alone. Thus,

so the matrix of the transformation is A_{6}=\(\left[\begin{array}{lll}

0 & 1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 1

\end{array}\right]\).

g. Suppose that a linear transformation L_{7} satisfies Find the matrix A_{7} of the transformation L_{7}. What is the geometric effect of this transformation?

Answer:

The matrix of this transformation is . The geometric effect of L_{7} is to stretch by a factor of 2 in the x-direction and scale by a factor of \(\frac{1}{2}\) in the z-direction.

h. Suppose that a linear transformation L_{8} satisfies . Find the matrix of the transformation L_{8}. What is the geometric effect of this transformation?

Answer:

The matrix of this transformation is . The geometric effect of L_{8} is to rotate by 45° in the xy-plane, to scale by \(\sqrt{2}\) in both the x and y directions, and to not change in the z-direction.

### Eureka Math Precalculus Module 2 Lesson 9 Exploratory Challenge Answer Key

Exploratory Challenge 1

Transformations L_{1}–L_{8} refer to the linear transformations from the Opening Exercise. For each pair:

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 L_{B} with matrix B and then L_{A} with matrix A, the matrix that corresponds to the composition L_{A}∘L_{B} is AB. That is, L_{B} is applied first, but matrix B 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?

a. Perform L_{6} and then L_{6}.

Answer:

i. Since L_{6} reflects across the plane through y=x that is perpendicular to the xy-plane, performing L_{6} twice in succession will result in the identity transformation.

ii. A_{6}⋅A_{6}=. Since A_{6}⋅A_{6} is the identity matrix, we know that

L_{6}∘L_{6} is the identity transformation.

iii. The conjecture was correct.

b. Perform L_{1} and then L_{2}.

Answer:

i. Sample student response: Since L_{1} rotates 90° about the z-axis and L_{2} rotates 90° about the y-axis, the composition L_{2}∘L_{1} should rotate 180° about the line y=-x in the xy-plane.

ii.

iii. The conjecture in part (i) is not correct. While it appears that the composition L_{2}∘L_{1} is a rotation by 180° about the line y=-x, it is not because, for example, point (0,1,0) is transformed to point (0,0,-1) and does not remain on the y-axis after the transformation. Thus, this cannot be a rotation around the y-axis.

c. Perform L_{4} and then L_{5}.

Answer:

i. Since L_{4} projects onto the xy-plane and L_{5} projects onto the xz-plane, the composition L_{5}∘L_{4} will project onto the x-axis.

ii.

iii. The conjecture in part (i) is correct. The cube is first transformed to a square in the xy-plane and then transformed onto a segment on the x-axis.

d. Perform L_{4} and then L_{3}.

Answer:

i. Since L_{4} projects onto the xy-plane and L_{3} scales in the x, y, and z directions, the composition L_{3}∘L_{4} will project onto the xy-plane and scale in the x and y directions.

ii.

iii. The conjecture in part (i) is correct.

e. Perform L_{3} and then L_{7}.

Answer:

i. Since L_{3} scales in the x, y, and z directions and so does L_{7}, the composition will be a transformation that also scales in all three directions but with different scale factors.

ii.

iii. The conjecture from (i) is correct.

f. Perform L_{8} and then L_{4}.

Answer:

i. Transformation L_{8} rotates the unit cube by 45° about the z-axis and stretches by a factor of \(\sqrt{2}\) in both the x and y directions, while L_{4} projects the image onto the xy-plane. The composition will transform the unit cube into a larger square that has been rotated 45° in the xy-plane.

ii.

iii. The conjecture from part (i) is correct.

g. Perform L_{4} and then L_{6}.

Answer:

i. Transformation L_{4} projects onto the xy-plane, and transformation L_{6} reflects across the plane through the line y=x in the xy-plane and is perpendicular to the xy-plane, so the composition L_{6}∘L_{4} will appear to be the reflection in the xy-plane across the line y=x.

ii.

iii. The conjecture in part (i) is correct.

h. Perform L_{2} and then L_{7}.

Answer:

i. Since L_{2} rotates 90° around the y-axis and L_{7} scales in the x and z directions, the composition L_{7}∘L_{2} will rotate and scale simultaneously.

ii.

iii. The conjecture in part (i) is correct.

i. Perform L_{8} and then L_{8}.

Answer:

i. Since L_{8} rotates by 45° about the z-axis and scales by \(\sqrt{2}\) in the x and y directions, performing this transformation twice will rotate by 90° about the z-axis and scale by 2 in the x and y directions.

ii.

iii. The conjecture in part (i) is correct.

Exploratory Challenge 2

Transformations L_{1}–L_{8} refer to the transformations from the Opening Exercise. For each of the following pairs of matrices A and B below, compare the transformations L_{A}∘L_{B} and L_{B}∘L_{A}.

a. L_{4} and L_{5}

Answer:

Transformation L_{4}∘L_{5} has matrix representation A_{4}⋅A_{5}= and transformation L_{5}∘L_{4} has matrix representation

Since the two transformations have the same matrix representation, they are the same transformation:

L_{5}∘L_{4}=L_{4}∘L_{5}

b. L_{2} and L_{5}

Answer:

Transformation L_{2} \(\text { o }\) L_{5} has matrix representation and

transformation L_{5} \(\text { o }\) L_{2} has matrix representation A_{5}∙A_{2} =

Since the two transformations have the same matrix representation, they are the same transformation:

L_{2} \(\text { o }\)L_{5} = L_{5} \(\text { o }\) L_{2}

c. L_{3} and L_{7}

Answer:

Transformation L_{3}∘L_{7} has matrix representation and transformation as matrix representation

Since the two transformations have the same matrix representation, they are the same transformation:

L_{3}\(\text { o }\)L_{7}=L_{7}\(\text { o }\)L_{3}

d. L_{3} and L_{6}

Answer:

Transformation L_{3}\(\text { o }\)L_{6} has matrix representation and transformation L_{6}\(\text { o }\)L_{3} has matrix representation

Since the two transformations have different matrix representations, they are not the same transformation: L_{3}∘L_{6}≠L_{6}∘L_{3}

e. L_{7} and L_{1}

Answer:

Transformation L_{7}\(\text { o }\)L_{1} has matrix representation and transformation L_{1}∘L_{7} has matrix representation

Since the two transformations have different matrix representations, they are not the same transformation: L_{1}∘L_{7}≠L_{7}∘L_{1}

f. What can you conclude about the order in which you compose two linear transformations?

Answer:

In some cases, the order of composition of two linear transformations matters. For two matrices A and B, the transformation L_{A}∘L_{B} is not always the same transformation as L_{B}∘L_{A}.

### Eureka Math Precalculus Module 2 Lesson 9 Problem Set Answer Key

Question 1.

Let A be the matrix representing a dilation of \(\frac{1}{2}\), and let B be the matrix representing a reflection across the

yz-plane.

a. Write A and B.

Answer:

b. Evaluate AB. What does this matrix represent?

Answer:

AB is a reflection across the yz-plane followed by a dilation of \(\frac{1}{2}\).

c. Let x=\(\left[\begin{array}{l}

5 \\

6 \\

4

\end{array}\right]\), y=\(\left[\begin{array}{l}

-1 \\

3 \\

2

\end{array}\right]\), and z=\(\left[\begin{array}{l}

8 \\

-2 \\

-4

\end{array}\right]\). Find (AB)x, (AB)y, and (AB)z.

Answer:

Question 2.

Let A be the matrix representing a rotation of 30° about the x-axis, and let B be the matrix representing a dilation of 5.

a. Write A and B.

Answer:

b. Evaluate AB. What does this matrix represent?

Answer:

AB is a dilation of 5 followed by a rotation of 30° about the x-axis.

c. Let x=\(\left[\begin{array}{l}

1 \\

0 \\

0

\end{array}\right]\), y=\(\left[\begin{array}{l}

0 \\

1 \\

0

\end{array}\right]\), z=\(\left[\begin{array}{l}

0 \\

0 \\

1

\end{array}\right]\). Find (AB)x, (AB)y, and (AB)z.

Answer:

Question 3.

Let A be the matrix representing a dilation of 3, and let B be the matrix representing a reflection across the plane y=x.

a. Write A and B.

Answer:

b. Evaluate AB. What does this matrix represent?

Answer:

AB=\(\left[\begin{array}{lll}

0 & 3 & 0 \\

3 & 0 & 0 \\

0 & 0 & 3

\end{array}\right]\)

AB is a reflection across the y=x plane followed by a dilation of 3.

c. Let x=\(\left[\begin{array}{c}

-2 \\

7 \\

3

\end{array}\right]\). Find (AB)x.

Answer:

(AB)x=\(\left[\begin{array}{c}

21 \\

-6 \\

9

\end{array}\right]\)

Question 4.

Let A=\(\left[\begin{array}{lll}

3 & 0 & 0 \\

3 & 3 & 0 \\

0 & 0 & 1

\end{array}\right]\), B=\(\left[\begin{array}{ccc}

0 & -1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 1

\end{array}\right]\).

a. Evaluate AB.

Answer:

\(\left[\begin{array}{ccc}

0 & -3 & 0 \\

3 & -3 & 0 \\

0 & 0 & 1

\end{array}\right]\)

b. Let x=\(\left[\begin{array}{c}

-2 \\

2 \\

5

\end{array}\right]\). Find (AB)x.

Answer:

\(\left[\begin{array}{c}

-6 \\

-12 \\

5

\end{array}\right]\)

c. Graph x and (AB)x.

Answer:

Question 5.

Let A=\(\left[\begin{array}{lll}

\frac{1}{3} & 0 & 0 \\

0 & 1 & 0 \\

2 & 0 & \frac{1}{3}

\end{array}\right]\), B=\(\left[\begin{array}{ccc}

3 & 1 & 0 \\

1 & -3 & 0 \\

0 & 0 & 1

\end{array}\right]\)

a. Evaluate AB.

Answer:

\(\left[\begin{array}{ccc}

1 & \frac{1}{3} & 0 \\

1 & -3 & 0 \\

6 & 2 & \frac{1}{3}

\end{array}\right]\)

b. Let x=\(\left[\begin{array}{l}

\mathbf{0} \\

3 \\

2

\end{array}\right]\). Find (AB)x.

Answer:

\(\left[\begin{array}{c}

1 \\

-9 \\

\frac{20}{3}

\end{array}\right]\)

c. Graph x and (AB)x.

Answer:

Question 6.

Let A=\(\left[\begin{array}{lll}

2 & 0 & 0 \\

0 & 3 & 0 \\

0 & 0 & 8

\end{array}\right]\), B=\(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 0

\end{array}\right]\).

a. Evaluate AB.

Answer:

\(\left[\begin{array}{lll}

2 & 0 & 0 \\

0 & 3 & 0 \\

0 & 0 & 0

\end{array}\right]\)

b. Let x=\(\left[\begin{array}{c}

1 \\

-2 \\

4

\end{array}\right]\). Find (AB)x.

Answer:

\(\left[\begin{array}{c}

2 \\

-6 \\

0

\end{array}\right]\)

c. Graph x and (AB)x.

Answer:

d. What does AB represent geometrically?

Answer:

AB represents a projection onto the xy-plane, and a dilation of 2 in the x-direction and 3 in the y-direction.

Question 7.

Let A, B, C be 3×3 matrices representing linear transformations.

a. What does A(BC) represent?

Answer:

The linear transformation of applying the linear transformation that C represents followed by the transformation that B represents followed by the transformation that A represents

b. Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left?

Answer:

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 (AB)C represent something different from A(BC)? Explain.

Answer:

No, it does not. This is the linear transformation obtained by applying C and then AB, which is B followed by A.

Question 8.

Let AB represent any composition of linear transformations in R^{3}. What is the value of (AB)x where x=\(\left[\begin{array}{l}

\mathbf{0} \\

\mathbf{0} \\

\mathbf{0}

\end{array}\right]\)?

Answer:

Since a composition of linear transformations in R^{3} is also a linear transformation, we know that applying it to the origin will result in no change.

### Eureka Math Precalculus Module 2 Lesson 9 Exit Ticket Answer Key

Let A be the matrix representing a rotation about the z-axis of 45° and B be the matrix representing a dilation of 2.

a. Write down A and B.

Answer:

b. Let x=\(\left[\begin{array}{c}

3 \\

-1 \\

2

\end{array}\right]\). Find the matrix representing a dilation of x by 2 followed by a rotation about the z-axis of 45°.

Answer:

AB=\(\left[\begin{array}{ccc}

\sqrt{2} & -\sqrt{2} & 0 \\

\sqrt{2} & \sqrt{2} & 0 \\

0 & 0 & 2

\end{array}\right]\)

c. Do your best to sketch a picture of x, x after the first transformation, and x after both transformations. You may use technology to help you.

Answer: