Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Lesson 6: Linear Transformations as Matrices
Student Outcomes

Students verify that 3 × 3 matrices represent linear transformations from ℝ3 → ℝ3 . Students investigate the
matrices associated with transformations such as dilations and reflections in a coordinate plane.

Students explore properties of vector arithmetic, including the commutative, associative, and distributive
properties.
Classwork
Opening Exercise (3 minutes)
In the Opening Exercise, students are reminded of their work with 2 × 2 matrices and prove that the matrix given
represents a linear transformation.
Opening Exercise
Let 𝑨 = (
𝒙𝟏
𝒚𝟏
𝟕 −𝟐
), 𝒙 = (𝒙 ), and 𝒚 = (𝒚 ). Does this represent a linear transformation? Explain how you know.
𝟓 −𝟑
𝟐
𝟐
A linear transformation satisfies the following conditions: 𝑳(𝒙 + 𝒚) = 𝑳(𝒙) + 𝑳(𝒚) and 𝑳(𝒌𝒙) = 𝒌𝑳(𝒙)
𝑨 ⋅ (𝒙 + 𝒚) = (
𝟕𝒙 + 𝟕𝒚𝟏 − 𝟐𝒙𝟐 − 𝟐𝒚𝟐
𝟕(𝒙𝟏 + 𝒚𝟏 ) − 𝟐(𝒙𝟐 + 𝒚𝟐 )
𝟕 −𝟐 𝒙𝟏 + 𝒚𝟏
)(
)=(
)=( 𝟏
)
𝟓𝒙𝟏 + 𝟓𝒚𝟏 − 𝟑𝒙𝟐 − 𝟑𝒚𝟐
𝟓 −𝟑 𝒙𝟐 + 𝒚𝟐
𝟓(𝒙𝟏 + 𝒚𝟏 ) − 𝟑(𝒙𝟐 + 𝒚𝟐 )
𝑨(𝒙) = (
MP.3
𝟕(𝒙𝟏 ) − 𝟐(𝒙𝟐 )
𝟕 −𝟐 𝒙𝟏
)( ) = (
)
𝟓 −𝟑 𝒙𝟐
𝟓(𝒙𝟏 ) − 𝟑(𝒙𝟐 )
𝟕(𝒚𝟏 ) − 𝟐(𝒚𝟐 )
𝟕 −𝟐 𝒚𝟏
𝑨(𝒚) = (
)( ) = (
)
𝟓 −𝟑 𝒚𝟐
𝟓(𝒚𝟏 ) − 𝟑(𝒚𝟐 )
𝟕(𝒚𝟏 ) − 𝟐(𝒚𝟐 )
𝟕(𝒙𝟏 ) − 𝟐(𝒙𝟐 )
𝟕(𝒙𝟏 ) − 𝟐(𝒙𝟐 ) +𝟕(𝒚𝟏 ) − 𝟐(𝒚𝟐 )
𝑨(𝒙) + 𝑨(𝒚) = (
)+(
)=(
)
𝟓(𝒙𝟏 ) − 𝟑(𝒙𝟐 )
𝟓(𝒚𝟏 ) − 𝟑(𝒚𝟐 )
𝟓(𝒙𝟏 ) − 𝟑(𝒙𝟐 ) +𝟓(𝒚𝟏 ) − 𝟑(𝒚𝟐 )
𝑨(𝒙 + 𝒚) = 𝑨(𝒙) + 𝑨(𝒚) by the distributive property
Discussion (7 minutes): Linear Transformations ℝ𝟑 → ℝ𝟑


In previous lessons, we have seen that 2 × 2 matrices represent linear transformations from ℝ2 → ℝ2 . Do you
think that a 3 × 3 matrix would represent a linear transformation from ℝ3 → ℝ3 ? Can you prove it?
𝑦1
𝑥1
7
−2 −7
Let 𝐴 = ( 5
−3 −6), and let 𝑥 = (𝑥2 ) and 𝑦 = (𝑦2 ) be points in ℝ3 . Show that 𝐴(𝑥 + 𝑦) = 𝐴𝑥 + 𝐴𝑦.
𝑥3
𝑦3
−10 −6 6

7
𝐴 ∙ (𝑥 + 𝑦) = ( 5
−10
Lesson 6:
−2
−3
−6
7(𝑥1 + 𝑦1 ) − 2(𝑥2 + 𝑦2 ) − 7(𝑥3 + 𝑦3 )
−7 𝑥1 + 𝑦1
−6) (𝑥2 + 𝑦2 ) = ( 5(𝑥1 + 𝑦1 ) − 3(𝑥2 + 𝑦2 ) − 6(𝑥3 + 𝑦3 ) )
𝑥3 + 𝑦3
6
−10(𝑥1 + 𝑦1 ) − 6(𝑥2 + 𝑦2 ) + 6(𝑥3 + 𝑦3 )
Linear Transformations as Matrices
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M2
PRECALCULUS AND ADVANCED TOPICS



7
𝐴𝑥 = ( 5
−10
−2
−3
−6
7𝑥1 − 2𝑥2 − 7𝑥3
−7 𝑥1
−6) (𝑥2 ) = ( 5𝑥1 − 3𝑥2 − 6𝑥3 )
𝑥3
−10𝑥1 − 6𝑥2 + 6𝑥3
6

7
𝐴𝑦 = ( 5
−10
−2
−3
−6
7𝑦1 − 2𝑦2 − 7𝑦3
−7 𝑦1
−6) (𝑦2 ) = ( 5𝑦1 − 3𝑦2 − 6𝑦3 )
𝑦3
−10𝑦1 − 6𝑦2 + 6𝑦3
6

𝐴𝑥 + 𝐴𝑦 = (

We can see that 𝐴 ∙ (𝑥 + 𝑦) = 𝐴𝑥 + 𝐴𝑦 by the distributive property.
Now show that 𝐴(𝑘 ∙ 𝑥) = 𝑘 ∙ 𝐴𝑥, where 𝑘 represents a real number.
7𝑘 ∙ 𝑥1 −2𝑘 ∙ 𝑥2 − 7𝑘 ∙ 𝑥3
7
−2 −7 𝑘 ∙ 𝑥1
 𝐴(𝑘 ∙ 𝑥) = ( 5
−3 −6) (𝑘 ∙ 𝑥2 ) = ( 5𝑘 ∙ 𝑥1 −3𝑘 ∙ 𝑥2 − 6𝑘 ∙ 𝑥3 )
𝑘 ∙ 𝑥3
−10𝑘 ∙ 𝑥1 −6𝑘 ∙ 𝑥2 + 6𝑘 ∙ 𝑥3
−10 −6 6
−2
−3
−6
7𝑥1 − 2𝑥2 − 7𝑥3
7𝑘 ∙ 𝑥1 −2𝑘 ∙ 𝑥2 − 7𝑘 ∙ 𝑥3
−7 𝑥1
−6) (𝑥2 ) = 𝑘 ∙ ( 5𝑥1 − 3𝑥2 − 6𝑥3 ) = ( 5𝑘 ∙ 𝑥1 −3𝑘 ∙ 𝑥2 − 6𝑘 ∙ 𝑥3 )
𝑥3
−10𝑥1 − 6𝑥2 + 6𝑥3
−10𝑘 ∙ 𝑥1 −6𝑘 ∙ 𝑥2 + 6𝑘 ∙ 𝑥3
6

7
𝑘 ∙ 𝐴𝑥 = 𝑘 ∙ ( 5
−10

We see that 𝐴(𝑘 ∙ 𝑥) = 𝑘 ∙ 𝐴𝑥.
Let’s briefly take a closer look at the reasoning used here. What properties of
arithmetic are involved in comparing, for example, 7 ∙ (𝑘 ∙ 𝑥1 ) with 𝑘 ∙ (7 ∙ 𝑥1 )?
In other words, how exactly do we know these are the same number?


7𝑥1 − 2𝑥2 − 7𝑥3 + 7𝑦1 − 2𝑦2 − 7𝑦3
5𝑥1 − 3𝑥2 − 6𝑥3 + 5𝑦1 − 3𝑦2 − 6𝑦3 )
−10𝑥1 − 6𝑥2 + 6𝑥3 + −10𝑦1 − 6𝑦2 + 6𝑦3
We can use the associative property and the commutative property to
know these are the same number.
Now we’ve proved that the transformation 𝐴(𝑥) does indeed represent a linear
transformation, at least for this particular 3 × 3 matrix. We could use exactly
the same reasoning to show that this is also true for any 3 × 3 matrix. In fact,
mathematicians showed in the 1800s that every linear transformation from
ℝ3 → ℝ3 can be represented by a 3 × 3 matrix! This is one reason why the
theory of matrices is so powerful.
Exploratory Challenge 1 (10 minutes): The Geometry of 3-D Matrix
Transformations
In this activity, students work in groups of four on each of the challenges. Monitor the
progress of the class as they work independently. For each challenge, select a particular
group to present their findings to the class at the conclusion of the activity. Allow
approximately 7 minutes for students to work in their groups and approximately 3 minutes
for students to present their findings.
Lesson 6:
Linear Transformations as Matrices
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Scaffolding:
 For students still struggling
with matrix multiplication,
give them a matrix with
blanks to complete so they
can see how many terms
they should have in each
row such as
___ + ___ + ___
(___ + ___ + ___).
Students would fill in the
entry for each blank. For
example,
1 3
2
(−5 2) ( ) =
−1
4 7
1 ∙ 2 + 3 ∙ −1
(−5 ∙ 2 + 2 ∙ −1).
4 ∙ 2 + 7 ∙ −1
 Advanced learners can
instead work with a
general 3 × 3 matrix as
linearity conditions are
explored. The reasoning is
exactly the same.
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
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?
(

Find a matrix 𝐴 such that 𝐴𝑥 = 𝑥 for each point 𝑥 in ℝ3 . Describe the geometric effect of this transformation.
How might we call such a matrix?
1 0 0
 We can choose 𝐴 = (0 1 0).
0 0 1
𝑥
𝑥1
1 ∙ 𝑥1 + 0 ∙ 𝑥2 + 0 ∙ 𝑥3
1 0 0
1
𝑥
𝑥
0
∙
𝑥
+
1
∙
𝑥
+
0
∙
𝑥
 𝐴𝑥 = (0 1 0) ( 2 ) = (
1
2
3) = ( 2)
𝑥3
0 ∙ 𝑥1 + 0 ∙ 𝑥2 + 1 ∙ 𝑥3
0 0 1 𝑥3
 The name “identity matrix” makes sense since the output is identical to the input.
b.
What matrix in ℝ𝟐 serves the role of 𝟎 in the real number system? What is that role?
(

MP.3
𝟏 𝟎
) serves the role of the multiplicative identity.
𝟎 𝟏
𝟎 𝟎
) serves the role of the additive identity.
𝟎 𝟎
Find a matrix 𝐴 such that 𝐴𝑥 = 0 for each point 𝑥 in ℝ3 . Describe the geometric effect of this transformation.
0 0 0 𝑥1
0
 (0 0 0) (𝑥2 ) = (0) This transformation takes every point in space and maps it to the origin.
0 0 0 𝑥3
0
c.
What is the result of scalar multiplication in ℝ𝟐 ?
Multiplying by a scalar, 𝒌, dilates each point in ℝ𝟐 by a factor of 𝒌.

Find a matrix 𝐴 such that the mapping 𝑥 ↦ 𝐴𝑥 dilates each point in ℝ3 by a factor of 3.
𝑥1
3𝑥1
3 0 0 𝑥1
 (0 3 0) (𝑥2 ) = (3𝑥2 ) = 3 (𝑥2 ) Thus, each point in ℝ3 is dilated by a factor of 3.
𝑥3
3𝑥3
0 0 3 𝑥3
d.
Given a complex number 𝒂 + 𝒃𝒊, what represents the transformation of that point across the real axis?
The conjugate, 𝒂 − 𝒃𝒊

1
Investigate the geometric effects of the transformation (0
0
or more specific examples.
1 0 0
5
5
 ( 0 1 0 ) ( 3) = ( 3 )
0 0 −1 4
−4
Lesson 6:
Linear Transformations as Matrices
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This file derived from PreCal-M2-TE-1.3.0-08.2015
0
1
0
𝑥1
0
0 ) (𝑥2 ). Illustrate your findings with one
−1 𝑥3
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS

This transformation induces a reflection across the 𝑥𝑦-plane.
z
MP.3
y
x
Exploratory Challenge 2 (18 minutes): Properties of Vector Arithmetic
In this activity, students work in groups of four on one of the challenges below. Assign different challenges to different
groups. When a group finishes one of the challenges, they may choose one of the other challenges to work on as time
allows.
Monitor the progress of the class as they work independently. For each challenge, select a particular group to present
their findings to the class at the conclusion of the activity.
For each challenge, students should explore part (a) from an algebraic point of view and part (b) from a geometric point
of view.
Allow approximately 10 minutes for students to work in their groups and organize their presentations. Allow
approximately 9 minutes for students to present their findings.
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 ?
We want to show that 𝒙 + 𝒚 = 𝒚 + 𝒙. First, let’s look at this problem algebraically.
𝒙𝟏
𝒚𝟏
𝒙 +𝒚
When we compute 𝒙 + 𝒚, we get (𝒙 ) + (𝒚 ) = (𝒙𝟏 + 𝒚𝟏 ).
𝟐
𝟐
𝟐
𝟐
𝒚𝟏
𝒙𝟏
𝒚 +𝒙
Now let’s compute 𝒚 + 𝒙. This time, we get (𝒚 ) + (𝒙 ) = (𝒚𝟏 + 𝒙𝟏 ).
𝟐
𝟐
𝟐
𝟐
The commutative property guarantees that the two components of these vectors are equal. For example,
𝒙𝟏 + 𝒚𝟏 = 𝒚𝟏 + 𝒙𝟏 . Since both components are equal, the vectors themselves must be equal. We can make a
similar argument to show that vector addition in ℝ𝟑 is commutative.
MP.3
𝟑
𝟓
Next, let’s see what all of this means geometrically. Let’s examine the points 𝒙 = ( ) and 𝒚 = ( ).
𝟏
𝟒
Lesson 6:
Linear Transformations as Matrices
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
𝟑
𝟓
Thinking in terms of translations, the sum 𝒙 + 𝒚 = ( ) + ( ) amounts to moving 𝟓 right and 𝟏 up, followed
𝟏
𝟒
by a movement that takes us 𝟑 right and 𝟒 up.
𝟑
𝟓
On the other hand, the sum 𝒚 + 𝒙 = ( ) + ( ) amounts to moving 𝟑 right and 𝟒 up, followed by a
𝟒
𝟏
movement that takes us 𝟓 right and 𝟏 up.
It should be clear that, in both cases, we’ve moved a total of 𝟖 units to the right and 𝟓 units up. So it makes
sense that, when viewed as translations, these two sums are the same.
Lastly, let’s consider vector addition in ℝ𝟑 from a geometric point of view.
MP.3
−𝟑
𝟓
Let’s consider the vectors (𝟒) and ( 𝟐 ). To show that addition is commutative, let’s imagine that Jack and
𝟑
𝟐
Jill are moving around a building. We’ll send them on two different journeys and see if they reach the same
destination.
−𝟑
𝟓
Jack’s movements will model the sum (𝟒) + ( 𝟐 ). He walks 𝟓 units east, 𝟒 units north, and then goes up 𝟐
𝟑
𝟐
flights of stairs. Next, he goes 𝟑 units west, 𝟐 units north, and then goes up 𝟑 flights of stairs.
−𝟑
𝟓
Jill starts at the same place as Jack. Her movements will model the sum ( 𝟐 ) + (𝟒). She walks 𝟑 units
𝟑
𝟐
west, 𝟐 units north, and then goes up 𝟑 flights of stairs. Then, she goes 𝟓 units east, 𝟒 units north, and then
climbs 𝟐 flights of stairs. It should be clear from this description that Jack and Jill both end up in a location
that is 𝟐 units east, 𝟔 units north, and 𝟓 stories above their starting point. In particular, they end up in the
same spot!
Lesson 6:
Linear Transformations as Matrices
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M2
PRECALCULUS AND ADVANCED TOPICS
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 ?
Let’s check to see if (𝒙 + 𝒚) + 𝒓 = 𝒙 + (𝒚 + 𝒓) for points in ℝ𝟐 .
𝒙
𝒚
𝒓
𝒓𝟏
𝒙 + 𝒚𝟏
(𝒙 + 𝒚) + 𝒓 = ((𝒙𝟏 ) + (𝒚𝟏 )) + (𝒓𝟏 ) = ( 𝟏
𝒙𝟐 + 𝒚𝟐 ) + (𝒓𝟐 )
𝟐
𝟐
𝟐
𝒙𝟏
𝒚𝟏
𝒓𝟏
𝒙𝟏
𝒚 +𝒓
𝒙 + (𝒚 + 𝒓) = (𝒙 ) + ((𝒚 ) + (𝒓 )) = (𝒙 ) + (𝒚𝟏 + 𝒓𝟏 )
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
The associative property guarantees that each of the components is equal. Let’s look at the first coordinate.
For example, we have (𝒙𝟏 + 𝒚𝟏 ) + 𝒓𝟏 , which is indeed the same as 𝒙𝟏 + (𝒚𝟏 + 𝒓𝟏 ). We can make a similar
argument for vectors in ℝ𝟑 .
Next, let’s examine the problem from a geometric point of view.
MP.3
These pictures make it clear that these two sums should be the same. In both cases, the overall journey is
equivalent to following each of the three paths separately. Now let’s look at the three-dimensional case.
As in the two-dimensional case, we are simply reaching the same location in two different ways, both of
which are equivalent to following the three individual paths separately.
Lesson 6:
Linear Transformations as Matrices
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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 ?
We want to show that 𝒌 ∙ (𝒙 + 𝒚) = 𝒌𝒙 + 𝒌𝒚.
𝒙𝟏
𝒚𝟏
𝒙 +𝒚
𝒌 ∙ (𝒙𝟏 + 𝒚𝟏 )
We have 𝒌 ∙ (𝒙 + 𝒚) = 𝒌 ∙ ((𝒙 ) + (𝒚 )) = 𝒌 ∙ (𝒙𝟏 + 𝒚𝟏 ) = (
).
𝒌 ∙ (𝒙𝟐 + 𝒚𝟐 )
𝟐
𝟐
𝟐
𝟐
𝒙𝟏
𝒚𝟏
𝒌𝒚
𝒌𝒙 + 𝒌𝒚𝟏
𝒌𝒙
Next, we have 𝒌𝒙 + 𝒌𝒚 = 𝒌 (𝒙 ) + 𝒌 (𝒚 ) = ( 𝟏 ) + ( 𝟏 ) = ( 𝟏
).
𝒌𝒙𝟐
𝒌𝒚𝟐
𝒌𝒙𝟐 + 𝒌𝒚𝟐
𝟐
𝟐
The distributive property guarantees that each of the components is equal. For example,
𝒌 ∙ (𝒙𝟏 + 𝒚𝟏 ) = 𝒌𝒙𝟏 + 𝒌𝒚𝟏 . This means that the vectors themselves are equal. A similar argument can be
used to show that this property holds for vectors in ℝ𝟑 .
Now let’s examine this property geometrically.
Suppose that Jack walks to the point marked 𝒙 + 𝒚 and then walks that distance again, ending at the spot
marked 𝟐(𝒙 + 𝒚). Now suppose Jill walks to the spot marked 𝟐𝒙 and then follows the path labeled 𝟐𝒚. The
picture makes it clear that Jack and Jill end up at the same spot.
Next, let’s examine the three-dimensional case. Again, the picture makes it clear that the distributive
property holds.
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
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 ?
𝒙𝟏
𝒂𝟏
𝒙 +𝒂
We have 𝒙 + 𝒂 = (𝒙 ) + (𝒂 ) = (𝒙𝟏 + 𝒂𝟏 ).
𝟐
𝟐
𝟐
𝟐
𝒙𝟏
𝒙 +𝒂
If this sum is equal to 𝒙, then we have (𝒙𝟏 + 𝒂𝟏 ) = (𝒙 ). This means that 𝒙𝟏 + 𝒂𝟏 = 𝒙𝟏 , which implies that
𝟐
𝟐
𝟐
𝒂𝟏 = 𝟎. In the same way, we can show that 𝒂𝟐 = 𝟎.
𝟎
𝟎
Thus, the identity element for ℝ𝟐 is ( ). Similarly, the identity element for ℝ𝟑 is (𝟎).
𝟎
𝟎
𝟐
𝟎
Let’s think about this geometrically. ( ) + ( ) means to go to (𝟐, 𝟓) and translate by (𝟎, 𝟎). That is, we
𝟓
𝟎
don’t translate at all.
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?
𝒂𝟏
𝒃
𝒂 + 𝒃𝟏
𝒂 + 𝒃𝟏
𝟎
We have 𝒂 + 𝒃 = (𝒂 ) + ( 𝟏 ) = ( 𝟏
). If this sum is equal to 𝟎, then ( 𝟏
) = ( ), which means
𝒃𝟐
𝒂𝟐 + 𝒃𝟐
𝒂𝟐 + 𝒃𝟐
𝟎
𝟐
that 𝒂𝟏 + 𝒃𝟏 = 𝟎. We can conclude that 𝒃𝟏 = −𝒂𝟏. Similarly, 𝒃𝟐 = −𝒂𝟐.
𝒂𝟏
−𝒂𝟏
𝟑
−𝟑
Thus, the additive inverse of (𝒃 ) is (−𝒃 ). For instance, the additive inverse of ( ) is ( ). Geometrically,
𝟓
−𝟓
𝟏
𝟏
we see that these vectors have the same length but point in opposite directions. Thus, their sum is (𝟎, 𝟎).
This makes sense because if Jack walks toward a spot that is 𝟑 miles east and 𝟓 miles north of the origin and
then walks toward the spot that is 𝟑 miles west and 𝟓 miles south of the origin, then he’ll end up right back at
the origin!
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Closing (2 minutes)
Students should write a brief response to the following questions in their notebooks.

How can we represent linear transformations from ℝ3 → ℝ3 ?


These transformations can be represented using 3 × 3 matrices.
Which properties of real numbers are true for vectors in ℝ2 and ℝ3 ?

Vector addition is commutative and associative. There is a 0 vector, and every vector has an additive
inverse. Scalar multiplication distributes over vector addition.
Exit Ticket (5 minutes)
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Name
Date
Lesson 6: Linear Transformations as Matrices
Exit Ticket
1.
1
4
3
Given 𝑥 = ( ), 𝑦 = ( ), 𝑧 = ( ), and 𝑘 = −2
2
2
1
a. Verify the associative property holds: 𝑥 + (𝑦 + 𝑧) = (𝑥 + 𝑦) + 𝑧.
b.
2.
Verify the distributive property holds: 𝑘(𝑥 + 𝑦) = 𝑘𝑥 + 𝑘𝑦.
Describe the geometric effect of the transformation on the 3 × 3 identity matrix given by the following matrices.
a.
2
[0
0
0 0
2 0]
0 2
b.
−1
[0
0
c.
1
[0
0
0
−1
0
d.
1
[0
0
0
0]
1
Lesson 6:
Linear Transformations as Matrices
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This file derived from PreCal-M2-TE-1.3.0-08.2015
0
1
0
0
1
0
0
0]
1
0
0]
−1
114
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
Exit Ticket Sample Solutions
1.
𝟏
𝟒
𝟑
Given 𝒙 = ( ), 𝒚 = ( ), 𝒛 = ( ), and 𝒌 = −𝟐
𝟐
𝟐
𝟏
a.
Verify the associative property holds: 𝒙 + (𝒚 + 𝒛) = (𝒙 + 𝒚) + 𝒛.
𝟏
𝟒
𝟑
𝟏
𝟕
𝟖
𝒙 + (𝒚 + 𝒛) = ( ) + (( ) + ( )) = ( ) + ( ) = ( )
𝟐
𝟑
𝟓
𝟐
𝟏
𝟐
(𝒙 + 𝒚) + 𝒛 = ((𝟏) + (𝟒)) + (𝟑) = (𝟓) + (𝟑) = (𝟖)
𝟐
𝟓
𝟐
𝟏
𝟒
𝟏
b.
Verify the distributive property holds: 𝒌(𝒙 + 𝒚) = 𝒌𝒙 + 𝒌𝒚.
𝟏
𝟒
−𝟏𝟎
𝟓
𝒌(𝒙 + 𝒚) = −𝟐 (( ) + ( )) = −𝟐 ( ) = (
)
𝟐
−𝟖
𝟐
𝟒
𝟏
𝟒
−𝟐
−𝟖
−𝟏𝟎
𝒌𝒙 + 𝒌𝒚 = −𝟐 ( ) + (−𝟐) ( ) = ( ) + ( ) = (
)
𝟐
−𝟖
𝟐
−𝟒
−𝟒
2.
Describe the geometric effect of the transformation on the 𝟑 × 𝟑 identity matrix given by the following matrices.
a.
𝟐 𝟎 𝟎
[𝟎 𝟐 𝟎]
𝟎 𝟎 𝟐
It dilates the point by a factor of 𝟐.
b.
−𝟏 𝟎 𝟎
[ 𝟎 𝟏 𝟎]
𝟎 𝟎 𝟏
It reflects the point about the 𝒚𝒛-plane.
c.
𝟏 𝟎 𝟎
[𝟎 −𝟏 𝟎]
𝟎 𝟎 𝟏
It reflects the point about the 𝒙𝒛-plane.
d.
𝟏 𝟎 𝟎
[𝟎 𝟏 𝟎 ]
𝟎 𝟎 −𝟏
It reflects the point about the 𝒙𝒚-plane.
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
Problem Set Sample Solutions
1.
Show that the associative property, 𝒙 + (𝒚 + 𝒛) = (𝒙 + 𝒚) + 𝒛, holds for the following.
a.
𝒙=(
(
𝟑
−𝟒
−𝟏
𝟑
−𝟐
−𝟓
) + (( ) + ( )) = ( ) + ( ) = ( )
𝟐
𝟓
𝟓
−𝟐
−𝟐
𝟕
((
b.
𝟑
−𝟒
−𝟏
), 𝒚 = ( ), 𝒛 = ( )
𝟐
𝟓
−𝟐
𝟑
−𝟒
−𝟏
−𝟏
−𝟏
−𝟐
) + ( )) + ( ) = ( ) + ( ) = ( )
𝟐
𝟓
𝟎
𝟓
𝟓
−𝟐
𝟐
𝟎
𝟑
𝒙 = (−𝟐), 𝒚 = ( 𝟓 ), 𝒛 = ( 𝟎 )
−𝟑
𝟏
−𝟐
𝟐
𝟎
𝟑
𝟐
𝟑
𝟓
(−𝟐) + (( 𝟓 ) + ( 𝟎 )) = (−𝟐) + ( 𝟓 ) = ( 𝟑 )
−𝟑
−𝟓
𝟏
−𝟐
𝟏
−𝟒
𝟐
𝟎
𝟑
𝟐
𝟑
𝟓
((−𝟐) + ( 𝟓 )) + ( 𝟎 ) = ( 𝟑 ) + ( 𝟎 ) = ( 𝟑 )
−𝟑
−𝟑
𝟏
−𝟐
−𝟏
−𝟒
2.
Show that the distributive property, 𝒌(𝒙 + 𝒚) = 𝒌𝒙 + 𝒌𝒚, holds for the following.
a.
𝒙=(
−𝟐
𝟓
), 𝒚 = ( ), 𝒌 = −𝟐
−𝟑
𝟒
−𝟐 ((
−𝟐
𝟑
−𝟔
𝟓
) + ( )) = −𝟐 ( ) = ( )
−𝟑
𝟒
𝟏
−𝟐
−𝟐
−𝟏𝟎
𝟒
−𝟔
𝟓
−𝟐 ( ) + (−𝟐) ( ) = (
)+( ) =( )
𝟔
−𝟖
−𝟑
𝟒
−𝟐
b.
𝟑
−𝟒
𝒙 = (−𝟐), 𝒚 = ( 𝟔 ), 𝒌 = −𝟑
𝟓
−𝟕
𝟑
−𝟒
−𝟏
𝟑
−𝟑 ((−𝟐) + ( 𝟔 )) = −𝟑 ( 𝟒 ) = (−𝟏𝟐)
𝟓
−𝟕
𝟔
−𝟐
𝟑
−𝟒
−𝟗
𝟏𝟐
𝟑
−𝟑 (−𝟐) + (−𝟑) ( 𝟔 ) = ( 𝟔 ) + (−𝟏𝟖) = (−𝟏𝟐)
𝟓
−𝟕
−𝟏𝟓
𝟔
𝟐𝟏
3.
Compute the following.
a.
𝟏 𝟎 𝟐 𝟐
(𝟎 𝟏 𝟐) (𝟏)
𝟐 𝟐 𝟑 𝟑
𝟖
(𝟕)
𝟏𝟓
Lesson 6:
Linear Transformations as Matrices
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M2
PRECALCULUS AND ADVANCED TOPICS
b.
−𝟏 𝟐
𝟑
𝟏
(𝟑
𝟏 −𝟐) (−𝟐)
𝟏 −𝟐 𝟑
𝟏
−𝟐
(−𝟏)
𝟖
c.
𝟏 𝟎 𝟐 𝟐
(𝟎 𝟏 𝟎) (𝟏)
𝟐 𝟎 𝟑 𝟐
𝟔
(𝟏)
𝟏𝟎
4.
𝒂 𝒅 𝒈
𝟑
Let 𝒙 = (𝟏). Compute 𝑳(𝒙) = [𝒃 𝒆 𝒉] ∙ 𝒙, plot the points, and describe the geometric effect to 𝒙.
𝒄 𝒇 𝒊
𝟐
−𝟏 𝟎 𝟎
a.
[ 𝟎 𝟏 𝟎]
𝟎 𝟎 𝟏
−𝟑
( 𝟏 ) It is reflected about the 𝒚𝒛-plane.
𝟐
b.
𝟐 𝟎 𝟎
[𝟎 𝟐 𝟎]
𝟎 𝟎 𝟐
𝟔
(𝟐) It is dilated by a factor of 𝟐.
𝟒
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
c.
𝟎 𝟏 𝟎
[𝟏 𝟎 𝟎]
𝟎 𝟎 𝟏
𝟏
(𝟑) It is reflected about the vertical plane through
𝟐
the line 𝒚 = 𝒙 on the 𝒙𝒚-plane.
d.
𝟏 𝟎 𝟎
[𝟎 𝟎 𝟏]
𝟎 𝟏 𝟎
𝟑
(𝟐) It is reflected about the vertical plane through the line
𝟏
𝒚 = 𝒛 on the 𝒛𝒚-plane.
5.
𝟑
Let 𝒙 = (𝟏).
𝟐
𝟎 𝟎
a.
[𝟎 𝟎
𝟎 𝟎
𝒂 𝒅 𝒈
Compute 𝑳(𝒙) = [𝒃 𝒆 𝒉] ∙ 𝒙. Describe the geometric effect to 𝒙.
𝒄 𝒇 𝒊
𝟎
𝟎]
𝟎
𝟎
(𝟎) It is mapped to the origin.
𝟎
b.
𝟑 𝟎 𝟎
[𝟎 𝟑 𝟎]
𝟎 𝟎 𝟑
𝟗
(𝟑) It is dilated by a factor of 𝟑.
𝟔
Lesson 6:
Linear Transformations as Matrices
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
c.
−𝟏 𝟎
𝟎
[ 𝟎 −𝟏 𝟎 ]
𝟎
𝟎 −𝟏
−𝟑
(−𝟏) It is mapped to the opposite side of the origin on the same line that is equal distance from the origin.
−𝟐
d.
−𝟏 𝟎 𝟎
[ 𝟎 𝟏 𝟎]
𝟎 𝟎 𝟏
−𝟑
( 𝟏 ) It is reflected about the 𝒚𝒛-plane.
𝟐
e.
𝟏 𝟎 𝟎
[𝟎 −𝟏 𝟎]
𝟎 𝟎 𝟏
𝟑
(−𝟏) It is reflected about the 𝒙𝒛-plane.
𝟐
f.
𝟏 𝟎 𝟎
[𝟎 𝟏 𝟎 ]
𝟎 𝟎 −𝟏
𝟑
( 𝟏 ) It is reflected about the 𝒙𝒚-plane.
−𝟐
g.
𝟎 𝟏 𝟎
[𝟏 𝟎 𝟎]
𝟎 𝟎 𝟏
𝟏
(𝟑) It is reflected about the vertical plane through the line 𝒚 = 𝒙 on the 𝒙𝒚-plane.
𝟐
h.
𝟏 𝟎 𝟎
[𝟎 𝟎 𝟏]
𝟎 𝟏 𝟎
𝟑
(𝟐) It is reflected about the vertical plane through the line 𝒚 = 𝒛 on the 𝒚𝒛-plane.
𝟏
i.
𝟎 𝟎 𝟏
[𝟎 𝟏 𝟎]
𝟏 𝟎 𝟎
𝟐
(𝟏) It is reflected about the vertical plane through the line 𝒙 = 𝒛 on the 𝒙𝒛-plane.
𝟑
Lesson 6:
Linear Transformations as Matrices
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This file derived from PreCal-M2-TE-1.3.0-08.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
PRECALCULUS AND ADVANCED TOPICS
6.
𝟏
Find the matrix that will transform the point 𝒙 = (𝟑) to the following point:
𝟐
−𝟒
a.
(−𝟏𝟐)
−𝟖
−𝟒 𝟎
𝟎
[ 𝟎 −𝟒 𝟎 ]
𝟎
𝟎 −𝟒
b.
𝟑
(𝟏)
𝟐
𝟎 𝟏 𝟎
[𝟏 𝟎 𝟎]
𝟎 𝟎 𝟏
7.
𝟐
Find the matrix/matrices that will transform the point 𝐱 = (𝟑) to the following point:
𝟏
a.
𝟔
𝐱 ′ = (𝟒)
𝟐
𝟎 𝟐 𝟎
[𝟐 𝟎 𝟎]
𝟎 𝟎 𝟐
Lesson 6:
b.
−𝟏
𝐱′ = ( 𝟑 )
𝟐
𝟎 𝟎 −𝟏
[𝟎 𝟏 𝟎 ]
𝟏 𝟎 𝟎
Linear Transformations as Matrices
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This file derived from PreCal-M2-TE-1.3.0-08.2015
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