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MATH 211 Winter 2013

Lecture Notes(Adapted by permission of K. Seyffarth)

Sections 2.6

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2.6 Matrix Transformations

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Linear Transformations

Definition

A transformation T : Rn

Rm

is a linear transformation if and only if,for all X,Y Rn and all scalars a,T1. T(X + Y) = T(X) + T(Y) (preservation of addition)

T2. T(aX) = aT(X) (preservation of scalar multiplication)

As a consequence of T2, for any linear transformation T,

T(0X) = 0T(X), implying T(0) = 0,

andT((1)X) = (1)T(X), implying T(X) = T(X),

i.e., T preserves the zero vector and T preserves the negative of a vector.

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Furthermore, if X1,X2, . . . ,Xk are vectors in Rn and Y is a linear

combination of X1,X2, . . . ,Xk, i.e.,

Y = a1X1 + a2X2 + + akXkfor some a1, a2, . . . , ak

R, then T1 and T2 used repeatedly give us

T(Y) = T(a1X1 + a2X2 + + akXk)= a1T(X1) + a2T(X2) + + akT(Xk),

i.e.,T preserves linear combinations.

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Example

Let T : R4 R3 be a linear transformation such that

T

1

10

2

= 2

31

and T0

111

= 5

01

. Find T1

324

.The only way it is possible to solve this problem is if

1

324

is a linear combination of

1

10

2

and

0

111

,

i.e., if there exist a, b R so that

13

24

= a

110

2

+ b

01

11

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Example (continued)

Solve the system of four equations in two variables:

1 0 11 1 30 1 2

2 1

4

1 0 10 1 20 0 00 0 0

Thus a = 1, b = 2, and

13

24

=

11

02

2

0

1

11

.

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Example (continued)

It follows that

T

13

24

= T

110

2

2

01

11

= T

110

2

2T

01

11

=

231

2 50

1

= 83

3

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Example (2.6 Example 2)Every matrix transformation is a linear transformation.

Proof.

Suppose T : Rn Rm is a matrix transformation induced by the m nmatrix A, i.e., T(X) = AX for each X Rn.Let X,Y

Rn and let a

R. Then

T(X + Y) = A(X + Y) = AX + AY = T(X) + T(Y),

proving that T preserves addition. Also,

T(aX) = A(aX) = a(AX) = aT(X),

proving that T preserves scalar multiplication.

Since T1 and T2 are satisfied, T is, in fact, a linear transformation.

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It turns out that the converse of this statement is also true, i.e., everylinear transformation from Rn to Rm is a matrix transformation.

Theorem (2.6 Theorem 2)Let T : Rn Rm be a transformation.

1 T is linear if and only if T is a matrix transformation.

2 If T is linear, then T is induced by the unique matrix

A =

T(E1) T(E2) T(En),

where Ej is the jth column of In.

Why does this work for finding A?

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The uniqueness in Theorem 2 guarantees that there is exactly one matrixfor any linear transformation, so it makes sense to say the matrix of a

linear transformation.

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Examples

Q0 : R2

R

2 is reflection in the x-axis, i.e.,

Q0

x

y

=

x

y.

We saw earlier that Q0 is induced by the matrix A = 1 00 1 , so

Q0 is a linear transformation.

By Theorem 2, Q0 is induced by the matrix

Q0(E1) Q0(E2)

=

Q0

10

Q0

01

=

1 00 1

= A.

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Examples (continued)

R2

: R2

R

2 is (counterclockwise) rotation about the origin through

an angle of 2 , and

R2

x

y

=

yx

.

We saw earlier that R2

is induced by the matrix B = 0 1

1 0, so

R2

is a linear transformation.

By Theorem 2, R2

is induced by the matrix

R

2(E1) R2 (E2)

=

R

2

10

R

2

01

=

0 11 0

= B.

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Example (2.6 Example 4)Q1 : R

2 R2 is reflection in the line y = x, i.e.,

Q1

a

b

=

b

a

.

Notice that

ba

=

0 11 0

ab

,

so Q1 is a matrix transformation, and hence a linear transformation.

Again, illustrating Theorem 2, Q1 is induced by the matrix

Q1(E1) Q1(E2)

=

Q1

10

Q1

01

=

0 11 0

.

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Example

Let T : R2 R3 be a transformation defined by

T

x

y

=

2xy

x + 2y

.

Show that T is a linear transformation.Solution. If T were a linear transformation, then T would be induced bythe matrix

A =

T(E1) T(E2)

=

T 1

0

T 0

1

= 2 0

0 11 2.

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Example

Let T : R2 R3 be a transformation defined by

T

x

y

=

2xy

x + 2y

.

Show that T is a linear transformation.Solution. If T were a linear transformation, then T would be induced bythe matrix

A =

T(E1) T(E2)

=

T 1

0

T 0

1

= 2 0

0 11 2.

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Example (continued)

Since

A x

y

=

2 0

0 11 2 x

y

=

2x

yx + 2y

= T x

y,

T is a matrix transformation, and therefore a linear transformation.

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Example

Let T : R2 R2 be a transformation defined by

T

xy

=

xy

x + y

.

Is T is a linear transformation? Explain.

Solution.

IfT

were a linear transformation, thenT

would be induced bythe matrix

A =

T(E1) T(E2)

=

T

10

T

01

=

0 01 1

.

Now

A

x

y

=

0 01 1

x

y

=

0

x + y

.

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Example (continued)

In particular,A

11

=

0 01 1

11

=

02

,

while

T 1

1

= 1

2.

Since A

11

= T

11

, T is not a linear transformation.

There is an alternate way to show that T is not a linear transformation.

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Example (continued)

Notice that

11

=

10

+

01

, and

T

11

=

12

,T

10

=

01

,T

01

=

01

.

From this we see that

T

11

= T

10

+ T

01

,

i.e., T does not preserve addition, and so T is not a linear transformation.

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Definition (Vector scalar multiplication)Let X R2 and let k R. Then kX is the vector in R2 that is |k| timesthe length of X; kX points the same directions as X if k > 0, andopposite to X if k < 0.

Definition (Vector addition)

Let X,Y R2, and consider the parallelogram defined by 0, X and Y.The vector X + Y corresponds to the fourth vertex of this parallelogram.

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R t ti i R2

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Rotation in R2

Let R :R

2

R

2

denote counterclockwise rotation about the originthrough and angle of, and let X,Y R2 and k R. Then

R(kX) = kR(X),

andR(X + Y) = R(X) + R(Y).

This means that R is a linear transformation, and hence a matrixtransformation.

Therefore, by our earlier theorem, we can find the matrix that induces Rby simply finding R(E1) and R(E2).

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R(E1) = R 10 = cos

sin ,

and

R(E2) = R

01

=

sin cos

Theorem (2.6 Theorem 4)The transformation R : R

2 R2 is a linear transformation, and isinduced by the matrix

cos sin

sin cos .

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Reflectio i R2

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Reflection in R2

Let Qm :R

2

R

2

denote reflection in the line y = mx, and letX,Y R2 and k R. Then

Qm(kX) = kQm(X),

andQm(X + Y) = Qm(X) + Qm(Y).

This means that Qm is a linear transformation and hence a matrixtransformation.

Therefore, by our earlier theorem, we can find the matrix that induces Qmby simply finding Qm(E1) and Qm(E2).

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However, an easier way to obtain the matrix for Qm is to use the followingobservation:

Qm =

R Q

0 R,

where is the angle between y = mx and the positive x axis.

Using the standard trigonometric identities:

cos = 11 + m2

and sin = m1 + m2

,

the matrix for Qm can be found by computing the product

cos

sin

sin cos

1 00 1

cos(

) sin(

)sin() cos()

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Theorem (2 6 Theorem 5)

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Theorem (2.6 Theorem 5)The transformation Qm : R

2 R2, denoting reflection in the line y = mx,is a linear transformation and is induced by the matrix

11 + m2

1 m2 2m

2m m2 1.

For reflection in the x-axis, m = 0, and the theorem yields the expected

matrix 1 00 1

.

However, the y-axis has undefined slope, so the theorem does not apply.

We use QY to denote reflection in the y-axis, and as weve already seen,QY is induced by the matrix 1 0

0 1

,

and thus is a linear transformation.Sections 2.6 Page 25/1

E l

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Example

Find the rotation or reflection that equals reflection in the x-axis followedby rotation through an angle of 2 .

We must find the matrix for the transformation R2

Q0.

Q0 is induced by A =

1 00 1

, and R

2is induced by

B =

cos 2 sin 2sin 2 cos

2

=

0 11 0

R2

Q0 is induced by

BA =

0 11 0

1 00 1

=

0 11 0

.

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Example (continued)

Notice that BA =

0 11 0

is a reflection matrix.

How do we know this?

Now, since 1 m2 = 0, we know that m = 1 or m = 1. But2m1+m2

= 1 > 0, so m > 0, implying m = 1.

Therefore,R

2

Q0 = Q1,

reflection in the line y = x.

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In general,

The composite of two reflections is a rotation.

The composite of two rotations is a rotation.

The composite of a reflection and a rotation is a reflection.

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Example

Find the rotation or reflection that equals reflection in the line y = xfollowed by reflection in the y-axis.

We must find the matrix for the transformation QY Q1.

Q1 is induced by

A = 12

0 2

2 0

=

0 11 0

,

and QY is induced by

B =

1 0

0 1.

Therefore, QY Q1 is induced by BA.

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Example (continued)

BA=

1 0

0 1 0

1

1 0 = 0 1

1 0 .

What transformation does BA induce?

Rotation through an angle such that

cos = 0 and sin = 1.

Therefore, QY Q1 = R2

= R32

.

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