TBIL-LA Standard Matrices (AT2)

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Section 3.2 Standard Matrices (AT2)

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Learning Outcomes

Translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.

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Subsection 3.2.1 Warm Up

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Remark 3.2.1.

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Recall that a linear map

T : V \to W

satisfies

$T (\vec{v} + \vec{w}) = T (\vec{v}) + T (\vec{w})$ for any $\vec{v}, \vec{w} \in V .$
$T (c \vec{v}) = c T (\vec{v})$ for any $c \in R, \vec{v} \in V .$

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In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.

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Activity 3.2.2.

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Can you recall the following?

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(a)

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Given a transformation, what do the terms domain and codomain mean?

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(b)

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What does the notation

T : V \to W

mean?

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Subsection 3.2.2 Class Activities

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Activity 3.2.3.

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Suppose

T : R^{3} \to R^{2}

is a linear map, and you know

T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}]

and

T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 3 \\ 2 \end{matrix}] .

What is

T ([\begin{matrix} 3 \\ 0 \\ 0 \end{matrix}]) ?

$[\begin{matrix} 6 \\ 3 \end{matrix}]$
$[\begin{matrix} - 9 \\ 6 \end{matrix}]$
$[\begin{matrix} - 4 \\ - 2 \end{matrix}]$
$[\begin{matrix} 6 \\ - 4 \end{matrix}]$

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Activity 3.2.4.

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Suppose

T : R^{3} \to R^{2}

is a linear map, and you know

T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}]

and

T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 3 \\ 2 \end{matrix}] .

What is

T ([\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]) ?

$[\begin{matrix} 2 \\ 1 \end{matrix}]$
$[\begin{matrix} 3 \\ - 1 \end{matrix}]$
$[\begin{matrix} - 1 \\ 3 \end{matrix}]$
$[\begin{matrix} 5 \\ - 8 \end{matrix}]$

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Activity 3.2.5.

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Suppose

T : R^{3} \to R^{2}

is a linear map, and you know

T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}]

and

T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 3 \\ 2 \end{matrix}] .

What is

T ([\begin{matrix} - 2 \\ 0 \\ - 3 \end{matrix}]) ?

$[\begin{matrix} 2 \\ 1 \end{matrix}]$
$[\begin{matrix} 3 \\ - 1 \end{matrix}]$
$[\begin{matrix} - 1 \\ 3 \end{matrix}]$
$[\begin{matrix} 5 \\ - 8 \end{matrix}]$

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Activity 3.2.6.

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Suppose

T : R^{3} \to R^{2}

is a linear map, and you know

T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}]

and

T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 3 \\ 2 \end{matrix}] .

What piece of information would help you compute

T ([\begin{matrix} 0 \\ 4 \\ - 1 \end{matrix}]) ?

The value of $T ([\begin{matrix} 0 \\ 4 \\ 0 \end{matrix}]) = [\begin{matrix} - 4 \\ 16 \end{matrix}] .$
The value of $T ([\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]) = [\begin{matrix} - 1 \\ 4 \end{matrix}] .$
The value of $T ([\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]) = [\begin{matrix} - 2 \\ 7 \end{matrix}] .$
Any of the above.

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Observation 3.2.7.

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Since all three choices in Activity 3.2.6 create a spanning and linearly independent set along with

[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]

and

[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}],

they each may be used to compute

T ([\begin{matrix} 0 \\ 4 \\ - 1 \end{matrix}]) :

T ([\begin{matrix} 0 \\ 4 \\ - 1 \end{matrix}]) = T ([\begin{matrix} 0 \\ 4 \\ 0 \end{matrix}]) - T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 4 \\ 16 \end{matrix}] - [\begin{matrix} - 3 \\ 2 \end{matrix}] = [\begin{matrix} - 1 \\ 14 \end{matrix}]

T ([\begin{matrix} 0 \\ 4 \\ - 1 \end{matrix}]) = 4 T ([\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]) - T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = 4 [\begin{matrix} - 1 \\ 4 \end{matrix}] - [\begin{matrix} - 3 \\ 2 \end{matrix}] = [\begin{matrix} - 1 \\ 14 \end{matrix}]

T ([\begin{matrix} 0 \\ 4 \\ - 1 \end{matrix}]) = 4 T ([\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]) - 5 T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) - 4 T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}])

= 4 [\begin{matrix} - 2 \\ 7 \end{matrix}] - 5 [\begin{matrix} - 3 \\ 2 \end{matrix}] - 4 [\begin{matrix} 2 \\ 1 \end{matrix}] = [\begin{matrix} - 8 + 15 - 8 \\ 28 - 10 - 4 \end{matrix}] = [\begin{matrix} - 1 \\ 14 \end{matrix}]

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Fact 3.2.8.

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Consider any basis

{{\vec{b}}_{1}, \dots, {\vec{b}}_{n}}

for

V .

Since every vector

\vec{v}

can be written as a linear combination of basis vectors,

\vec{v} = x_{1} {\vec{b}}_{1} + \dots + x_{n} {\vec{b}}_{n},

we may compute

T (\vec{v})

as follows:

T (\vec{v}) = T (x_{1} {\vec{b}}_{1} + \dots + x_{n} {\vec{b}}_{n}) = x_{1} T ({\vec{b}}_{1}) + \dots + x_{n} T ({\vec{b}}_{n}) .

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Therefore any linear transformation

T : V \to W

can be defined by just describing the values of

T ({\vec{b}}_{i}) .

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Put another way, the images of the basis vectors completely determine the transformation

T .

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Definition 3.2.9.

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Since a linear transformation

T : R^{n} \to R^{m}

is determined by its action on the standard basis

{{\vec{e}}_{1}, \dots, {\vec{e}}_{n}},

it is convenient to store this information in an

m \times n

matrix, called the standard matrix of

T,

given by

[T ({\vec{e}}_{1}) \dots T ({\vec{e}}_{n})] .

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For example, let

T : R^{3} \to R^{2}

be the linear map determined by the following values for

T

applied to the standard basis of

R^{3} .

T ({\vec{e}}_{1}) = T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}] T ({\vec{e}}_{2}) = T ([\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]) = [\begin{matrix} - 1 \\ 4 \end{matrix}] T ({\vec{e}}_{3}) = T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 3 \\ 2 \end{matrix}]

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Then the standard matrix corresponding to

T

[\begin{array}{ccc} T ({\vec{e}}_{1}) & T ({\vec{e}}_{2}) & T ({\vec{e}}_{3}) \end{array}] = [\begin{array}{ccc} 2 & - 1 & - 3 \\ 1 & 4 & 2 \end{array}] .

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Activity 3.2.10.

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Let

T : R^{4} \to R^{3}

be the linear transformation given by

T ({\vec{e}}_{1}) = [\begin{matrix} 0 \\ 3 \\ - 2 \end{matrix}] T ({\vec{e}}_{2}) = [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}] T ({\vec{e}}_{3}) = [\begin{matrix} 4 \\ - 2 \\ 1 \end{matrix}] T ({\vec{e}}_{4}) = [\begin{matrix} 2 \\ 0 \\ 0 \end{matrix}]

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Write the standard matrix

[T ({\vec{e}}_{1}) \dots T ({\vec{e}}_{n})]

for

T .

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Activity 3.2.11.

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Let

T : R^{3} \to R^{2}

be the linear transformation given by

T ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} x + 3 z \\ 2 x - y - 4 z \end{matrix}]

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(a)

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Compute

T ({\vec{e}}_{1}),

T ({\vec{e}}_{2}),

and

T ({\vec{e}}_{3}) .

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(b)

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Find the standard matrix for

T .

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Fact 3.2.12.

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Because every linear map

T : R^{n} \to R^{m}

has a linear combination of the variables in each component, and thus

T ({\vec{e}}_{i})

yields exactly the coefficients of

x_{i},

the standard matrix for

T

is simply an array of the coefficients of the

x_{i} :

T ([\begin{matrix} x \\ y \\ z \\ w \end{matrix}]) = [\begin{matrix} a x + b y + c z + d w \\ e x + f y + g z + h w \end{matrix}] A = [\begin{array}{cccc} a & b & c & d \\ e & f & g & h \end{array}]

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Since the formula for a linear transformation

T

and its standard matrix

A

may both be used to compute the transformation of a vector

\vec{x},

we will often write

T (\vec{x})

and

A \vec{x}

interchangably:

T ([\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]) = [\begin{matrix} a x_{1} + b x_{2} + c x_{3} + d x_{4} \\ e x_{1} + f x_{2} + g x_{3} + h x_{4} \end{matrix}] = A [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{array}{cccc} a & b & c & d \\ e & f & g & h \end{array}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]

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Activity 3.2.13.

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Let

T : R^{4} \to R^{3}

be the linear transformation given by

T ({\vec{e}}_{1}) = [\begin{matrix} 0 \\ 3 \\ - 2 \end{matrix}] T ({\vec{e}}_{2}) = [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}] T ({\vec{e}}_{3}) = [\begin{matrix} 4 \\ - 2 \\ 1 \end{matrix}] T ({\vec{e}}_{4}) = [\begin{matrix} 2 \\ 0 \\ 0 \end{matrix}]

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Write the standard matrix

[T ({\vec{e}}_{1}) \dots T ({\vec{e}}_{n})]

for

T .

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Activity 3.2.14.

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(a)

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Explain and demonstrate how to compute the standard matrix for the linear transformation

S : R^{2} \to R^{4}

given by

S ([\begin{matrix} x_{1} \\ x_{2} \end{matrix}]) = [\begin{matrix} 9 x_{1} - 2 x_{2} \\ - 3 x_{1} \\ 5 x_{1} - x_{2} \\ - 6 x_{2} \end{matrix}]

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by computing transformations of the standard basic vectors:

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S ({\vec{e}}_{1}) = [\begin{matrix} ? \\ ? \\ ? \\ ? \end{matrix}] S ({\vec{e}}_{2}) = [\begin{matrix} ? \\ ? \\ ? \\ ? \end{matrix}] \to [\begin{array}{cc} ? & ? \\ ? & ? \\ ? & ? \\ ? & ? \end{array}]

Answer.

[\begin{array}{cc} 9 & - 2 \\ - 3 & 0 \\ 5 & - 1 \\ 0 & - 6 \end{array}]

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(b)

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Let

T : R^{4} \to R^{3}

be the linear transformation given by the standard matrix

[\begin{array}{cccc} - 2 & - 4 & 2 & - 2 \\ - 4 & 3 & - 3 & 2 \\ 5 & 0 & 2 & - 6 \end{array}] .

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Explain and demonstrate how to compute

T ([\begin{matrix} - 5 \\ 0 \\ - 3 \\ - 2 \end{matrix}])

by using the values of transformed standard basic vectors:

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T ([\begin{matrix} - 5 \\ 0 \\ - 3 \\ - 2 \end{matrix}]) = ? T ({\vec{e}}_{1}) + ? T ({\vec{e}}_{2}) + ? T ({\vec{e}}_{3}) + ? T ({\vec{e}}_{4})

Answer.

T ([\begin{matrix} - 5 \\ 0 \\ - 3 \\ - 2 \end{matrix}]) = [\begin{matrix} 8 \\ 25 \\ - 19 \end{matrix}]

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Subsection 3.2.3 Individual Practice

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Activity 3.2.15.

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Consider the linear transformation

R : R^{2} \to R^{2}

given by rotating vectors about the origin through an angle of

\frac{π}{4} = 45^{\circ} .

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(a)

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{\vec{e}}_{1}, {\vec{e}}_{2}

are the standard basis vectors of

R^{2},

calculate

R ({\vec{e}}_{1}), R ({\vec{e}}_{2}) .

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(b)

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What is the standard matrix representing

R ?

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Activity 3.2.16.

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Consider the linear transformation

S : R^{2} \to R^{2}

given by reflecting vectors across the line

x_{1} = x_{2} .

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(a)

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{\vec{e}}_{1}, {\vec{e}}_{2}

are the standard basis vectors of

R^{2},

calculate

S ({\vec{e}}_{1}), S ({\vec{e}}_{2}) .

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(b)

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What is the standard matrix representing

S ?

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Subsection 3.2.4 Videos

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Figure 23. Video: Using the standard matrix to compute the image of a vector

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Exercises 3.2.5 Exercises

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Exercises available at https://tbil.org/linear-algebra/preview/exercises/#/bank/AT2/.

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Subsection 3.2.6 Mathematical Writing Explorations

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We can represent images in the plane

R^{2}

using vectors, and manipulate those images with linear transformations. We introduce some notation in these explorations that is needed for their completion, but is not essential to the rest of the text. These have a geometric flair to them, and can be understood by thinking of geometric transformations in terms of standard matrices.

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Given two vectors

\vec{v} = [\begin{matrix} v_{1} \\ v_{2} \\ ⋮ \\ v_{n} \end{matrix}]

and

\vec{w} = [\begin{matrix} w_{1} \\ w_{2} \\ ⋮ \\ w_{n} \end{matrix}],

we define the dot product as

\vec{v} \cdot \vec{w} = v_{1} w_{1} + v_{2} w_{2} + \dots v_{n} w_{n} .

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Exploration 3.2.17.

For each of the following properties, determine if it is held by the dot product. Either provide a proof that the property holds, or provide a counter-example if it does not.

Distributive over addition (e.g., ( $\vec{u} + \vec{v}) \cdot \vec{w} = \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w}) ?$
Associative?
Commutative?

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Exploration 3.2.18.

Given the properties you proved in the last exploration, could the dot product take the place of

\oplus

as a vector space operation on

R^{n} ?

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Exploration 3.2.19.

Is the dot product a linear operator? That is, given vectors

\vec{u}, \vec{v}, \vec{w} \in R^{n},

and

k, m \in R,

is it true that

\vec{u} \cdot (k \vec{v} + m \vec{w}) = k (\vec{u} \cdot \vec{v}) + m (\vec{u} \cdot \vec{w}) .

Prove or provide a counter-example.

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Exploration 3.2.20.

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Assume

\vec{v} = [\begin{matrix} v_{1} \\ v_{2} \\ ⋮ \\ v_{n} \end{matrix}]

and define the length of a vector by

| \vec{v} | = {(\sum_{i = 1}^{n} v_{i}^{2})}^{1 / 2} .

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Prove that

| \vec{u} | = | \vec{v} |

if an only if

\vec{u} + \vec{v}

and

\vec{u} - \vec{v}

are perpendicular. You may use the fact (try and prove it!) that two vectors are perpendicular if and only if their dot product is zero.

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Exploration 3.2.21.

A dilation is given by by mapping a vector $\vec{v} = [\begin{matrix} x \\ y \end{matrix}]$ to some scalar multiple of $\vec{v} .$
A rotation is given by $\vec{v} \mapsto [\begin{matrix} \cos (θ) x - \sin (θ) y \\ \cos (θ) y + \sin (θ) x \end{matrix}] .$
A reflection of $\vec{v}$ over a line $l$ can be found by first finding a vector $\vec{l} = [\begin{matrix} l_{x} \\ l_{y} \end{matrix}]$ along $l,$ then $\vec{v} \mapsto 2 \frac{\vec{l} \cdot \vec{v}}{\vec{l} \cdot \vec{l}} \vec{l} - \vec{v} .$

Represent each of the following transformations with respect to the standard basis in

R^{2} .

Rotation through an angle $θ .$
Reflection over a line $l$ passing through the origin.
Dilation by some scalar $s .$

Prove that each transformation is linear, and that your matrix representations are correct.

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Subsection 3.2.7 Sample Problem and Solution

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Sample problem Example B.1.13.

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