TBIL-LA Linear Transformations (AT1)

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

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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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Definition 3.1.4.

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Given a linear transformation

T : V \to W,

V

is called the domain of

T

and

W

is called the co-domain of

T .

Figure 19. A linear transformation with a domain of $R^{3}$ and a co-domain of $R^{2}$

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Observation 3.1.5.

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One example of a linear transformation

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

is the projection of three-dimesional data onto a two-dimensional screen, as is necessary for computer animiation in film or video games.

Figure 20. A projection of a $3 D$ teapot onto a $2 D$ screen

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Activity 3.1.6.

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Let

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

be given by

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

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(a)

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Compute the result of adding vectors before a

T

transformation:

T ([\begin{matrix} x \\ y \\ z \end{matrix}] + [\begin{matrix} u \\ v \\ w \end{matrix}]) = T ([\begin{matrix} x + u \\ y + v \\ z + w \end{matrix}])

$[\begin{matrix} x - u + z - w \\ 3 y - 3 v \end{matrix}]$
$[\begin{matrix} x + u - z - w \\ 3 y + 3 v \end{matrix}]$
$[\begin{matrix} x + u \\ 3 y + 3 v \\ z + w \end{matrix}]$
$[\begin{matrix} x - u \\ 3 y - 3 v \\ z - w \end{matrix}]$

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(b)

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Compute the result of adding vectors after a

T

transformation:

T ([\begin{matrix} x \\ y \\ z \end{matrix}]) + T ([\begin{matrix} u \\ v \\ w \end{matrix}]) = [\begin{matrix} x - z \\ 3 y \end{matrix}] + [\begin{matrix} u - w \\ 3 v \end{matrix}]

$[\begin{matrix} x - u + z - w \\ 3 y - 3 v \end{matrix}]$
$[\begin{matrix} x + u - z - w \\ 3 y + 3 v \end{matrix}]$
$[\begin{matrix} x + u \\ 3 y + 3 v \\ z + w \end{matrix}]$
$[\begin{matrix} x - u \\ 3 y - 3 v \\ z - w \end{matrix}]$

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(c)

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T

a linear transformation?

Yes.
No.
More work is necessary to know.

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(d)

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Compute the result of scalar multiplcation before a

T

transformation:

T (c [\begin{matrix} x \\ y \\ z \end{matrix}]) = T ([\begin{matrix} c x \\ c y \\ c z \end{matrix}])

$[\begin{matrix} c x - c z \\ 3 c y \end{matrix}]$
$[\begin{matrix} c x + c z \\ - 3 c y \end{matrix}]$
$[\begin{matrix} x + c \\ 3 y + c \\ z + c \end{matrix}]$
$[\begin{matrix} x - c \\ 3 y - c \\ z - c \end{matrix}]$

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(e)

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Compute the result of scalar multiplcation after a

T

transformation:

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

$[\begin{matrix} c x - c z \\ 3 c y \end{matrix}]$
$[\begin{matrix} c x + c z \\ - 3 c y \end{matrix}]$
$[\begin{matrix} x + c \\ 3 y + c \\ z + c \end{matrix}]$
$[\begin{matrix} x - c \\ 3 y - c \\ z - c \end{matrix}]$

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(f)

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T

a linear transformation?

Yes.
No.
More work is necessary to know.

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Activity 3.1.7.

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Let

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

be given by

S ([\begin{matrix} x \\ y \end{matrix}]) = [\begin{matrix} x + y \\ x^{2} \\ y + 3 \\ y - 2^{x} \end{matrix}]

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(a)

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Compute

S ([\begin{matrix} 0 \\ 1 \end{matrix}] + [\begin{matrix} 2 \\ 3 \end{matrix}]) = S ([\begin{matrix} 2 \\ 4 \end{matrix}])

$[\begin{matrix} 6 \\ 4 \\ 7 \\ 0 \end{matrix}]$
$[\begin{matrix} - 3 \\ 0 \\ 1 \\ 5 \end{matrix}]$
$[\begin{matrix} - 3 \\ - 1 \\ 7 \\ 5 \end{matrix}]$
$[\begin{matrix} 6 \\ 4 \\ 10 \\ - 1 \end{matrix}]$

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(b)

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Compute

S ([\begin{matrix} 0 \\ 1 \end{matrix}]) + S ([\begin{matrix} 2 \\ 3 \end{matrix}]) = [\begin{matrix} 0 + 1 \\ 0^{2} \\ 1 + 3 \\ 1 - 2^{0} \end{matrix}] + [\begin{matrix} 2 + 3 \\ 2^{2} \\ 3 + 3 \\ 3 - 2^{2} \end{matrix}]

$[\begin{matrix} 6 \\ 4 \\ 7 \\ 0 \end{matrix}]$
$[\begin{matrix} - 3 \\ 0 \\ 1 \\ 5 \end{matrix}]$
$[\begin{matrix} - 3 \\ - 1 \\ 7 \\ 5 \end{matrix}]$
$[\begin{matrix} 6 \\ 4 \\ 10 \\ - 1 \end{matrix}]$

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(c)

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S

a linear transformation?

Yes.
No.
More work is necessary to know.

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Activity 3.1.8.

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Fill in the

?

s, assuming

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

is linear:

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

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Remark 3.1.9.

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In summary, any one of the following is enough to prove that

T : V \to W

is not a linear transformation:

Find specific values for $\vec{v}, \vec{w} \in V$ such that $T (\vec{v} + \vec{w}) \neq T (\vec{v}) + T (\vec{w}) .$
Find specific values for $\vec{v} \in V$ and $c \in R$ such that $T (c \vec{v}) \neq c T (\vec{v}) .$
Show $T (\vec{0}) \neq \vec{0} .$

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If you cannot do any of these, then

T

can be proven to be a linear transformation by doing both of the following:

For all $\vec{v}, \vec{w} \in V$ (not just specific values), $T (\vec{v} + \vec{w}) = T (\vec{v}) + T (\vec{w}) .$
For all $\vec{v} \in V$ and $c \in R$ (not just specific values), $T (c \vec{v}) = c T (\vec{v}) .$

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(Note the similarities between this process and showing that a subset of a vector space is or is not a subspace: Remark 2.3.11.)

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Activity 3.1.10.

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(a)

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Consider the following maps of Euclidean vectors

P : R^{3} \to R^{3}

and

Q : R^{3} \to R^{3}

defined by

P ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} - 2 x - 3 y - 3 z \\ 3 x + 4 y + 4 z \\ 3 x + 4 y + 5 z \end{matrix}] and Q ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} x - 4 y + 9 z \\ y - 2 z \\ 8 y^{2} - 3 x z \end{matrix}] .

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Which do you suspect?

$P$ is linear, but $Q$ is not.
$Q$ is linear, but $P$ is not.
Both maps are linear.
Neither map is linear.

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(b)

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Consider the following map of Euclidean vectors

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

S ([\begin{matrix} x \\ y \end{matrix}]) = [\begin{matrix} x + 2 y \\ 9 x y \end{matrix}] .

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Prove that

S

is not a linear transformation.

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(c)

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Consider the following map of Euclidean vectors

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

T ([\begin{matrix} x \\ y \end{matrix}]) = [\begin{matrix} 8 x - 6 y \\ 6 x - 4 y \end{matrix}] .

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Prove that

T

is a linear transformation.

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Subsection 3.1.3 Individual Practice

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Activity 3.1.11.

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Let

f (x) = x^{3} - 1 .

Then,

f : R \to R

is a function with domain and codomain equal to

R .

f (x)

is a linear transformation?

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Activity 3.1.12.

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(a)

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Is it the case that rotating

\vec{u} + \vec{v}

about the origin by

\frac{π}{2} = 90^{\circ}

is the same as first rotating each of

\vec{u}, \vec{v}

and then adding them together?

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(b)

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Is it the case that rotating

5 \vec{u}

about the origin by

\frac{π}{2} = 90^{\circ}

is the same as first rotating

\vec{u}

\frac{π}{2} = 90^{\circ}

and then scaling by

5 ?

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(c)

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Based on this, do you suspect that the transformation

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

given by rotating vectors about the origin through an angle of

\frac{π}{2} = 90^{\circ}

is linear? Do you think there is anything special about the angle

\frac{π}{2} = 90^{\circ} ?

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Activity 3.1.13.

In Activity 2.2.1, we made an analogy between vectors and linear combinations with ingredients and recipes. Let us think of cooking as a transformation of ingredients. In this analogy, would it be appropriate for us to consider "cooking" to be a linear transformation or not? Describe your reasoning.

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