2.2 Linear transformations

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An n{dimensional column vector is an n £ 1 matrix over F. The collection

of all n{dimensional column vectors is denoted by Fn.

Every matrix is associated with an important type of function called a

linear transformation.

DEFINITION 2.2.1 (Linear transformation) With A 2 Mm£n(F), we

associate the function TA : Fn ! Fm de¯ned by TA(X) = AX for all

X 2 Fn. More explicitly, using components, the above function takes the

form

y1 = a11x1 + a12x2 + ¢ ¢ ¢ + a1nxn

y2 = a21x1 + a22x2 + ¢ ¢ ¢ + a2nxn

...

ym = am1x1 + am2x2 + ¢ ¢ ¢ + amnxn;

where y1; y2; ¢ ¢ ¢ ; ym are the components of the column vector TA(X).

The function just de¯ned has the property that

TA(sX + tY ) = sTA(X) + tTA(Y ) (2.1)

for all s; t 2 F and all n{dimensional column vectors X; Y . For

TA(sX + tY ) = A(sX + tY ) = s(AX) + t(AY ) = sTA(X) + tTA(Y ):

28 CHAPTER 2. MATRICES

REMARK 2.2.1 It is easy to prove that if T : Fn ! Fm is a function

satisfying equation 2.1, then T = TA, where A is the m £ n matrix whose

columns are T(E1); : : : ; T(En), respectively, where E1; : : : ;En are the n{

dimensional unit vectors de¯ned by

E1 =

2

6664

1

0

...

0

3

7775

; : : : ; En =

2

6664

0

0

...

1

3

7775

:

One well{known example of a linear transformation arises from rotating

the (x; y){plane in 2-dimensional Euclidean space, anticlockwise through µ

radians. Here a point (x; y) will be transformed into the point (x1; y1),

where

x1 = x cos µ ¡ y sin µ

y1 = x sin µ + y cos µ:

In 3{dimensional Euclidean space, the equations

x1 = x cos µ ¡ y sin µ; y1 = x sin µ + y cos µ; z1 = z;

x1 = x; y1 = y cos Á ¡ z sin Á; z1 = y sin Á + z cos Á;

x1 = x cos à ¡ z sin Ã; y1 = y; z1 = x sin à + z cos Ã;

correspond to rotations about the positive z; x; y{axes, anticlockwise through

µ; Á; Ã radians, respectively.

The product of two matrices is related to the product of the correspond-

ing linear transformations:

If A is m£n and B is n£p, then the function TATB : Fp ! Fm, obtained

by ¯rst performing TB, then TA is in fact equal to the linear transformation

TAB. For if X 2 Fp, we have

TATB(X) = A(BX) = (AB)X = TAB(X):

The following example is useful for producing rotations in 3{dimensional

animated design. (See [27, pages 97{112].)

EXAMPLE 2.2.1 The linear transformation resulting from successively

rotating 3{dimensional space about the positive z; x; y{axes, anticlockwise

through µ; Á; Ã radians respectively, is equal to TABC, where

2.2. LINEAR TRANSFORMATIONS 29

µ

l

(x; y)

(x1; y1)

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

@

@

@@

@

@

@

Figure 2.1: Re°ection in a line.

C =

2

4

cos µ ¡sin µ 0

sin µ cos µ 0

0 0 1

3

5, B =

2

4

1 0 0

0 cos Á ¡sin Á

0 sin Á cos Á

3

5.

A =

2

4

cos à 0 ¡sin Ã

0 1 0

sin à 0 cos Ã

3

5.

The matrix ABC is quite complicated:

A(BC) =

2

4

cos à 0 ¡sin Ã

0 1 0

sin à 0 cos Ã

3

5

2

4

cos µ ¡sin µ 0

cos Á sin µ cos Á cos µ ¡sin Á

sin Á sin µ sin Á cos µ cos Á

3

5

=

2

4

cos à cos µ ¡ sin à sin Á sin µ ¡cos à sin µ ¡ sin à sin Á sin µ ¡sin à cos Á

cos Á sin µ cos Á cos µ ¡sin Á

sin à cos µ + cos à sin Á sin µ ¡sin à sin µ + cos à sin Á cos µ cos à cos Á

3

5.

EXAMPLE 2.2.2 Another example of a linear transformation arising from

geometry is re°ection of the plane in a line l inclined at an angle µ to the

positive x{axis.

We reduce the problem to the simpler case µ = 0, where the equations

of transformation are x1 = x; y1 = ¡y. First rotate the plane clockwise

through µ radians, thereby taking l into the x{axis; next re°ect the plane in

the x{axis; then rotate the plane anticlockwise through µ radians, thereby

restoring l to its original position.

30 CHAPTER 2. MATRICES

µ

l

(x; y)

(x1; y1)

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

¡

@

@

@

Figure 2.2: Projection on a line.

In terms of matrices, we get transformation equations

·

x1

y1

¸

=

·

cos µ ¡sin µ

sin µ cos µ

¸ ·

1 0

0 ¡1

¸ ·

cos (¡µ) ¡sin (¡µ)

sin (¡µ) cos (¡µ)

¸ ·

x

y

¸

=

·

cos µ sin µ

sin µ ¡cos µ

¸ ·

cos µ sin µ

¡sin µ cos µ

¸ ·

x

y

¸

=

·

cos 2µ sin 2µ

sin 2µ ¡cos 2µ

¸ ·

x

y

¸

:

The more general transformation

·

x1

y1

¸

= a

·

cos µ ¡sin µ

sin µ cos µ

¸ ·

x

y

¸

+

·

u

v

¸

; a > 0;

represents a rotation, followed by a scaling and then by a translation. Such

transformations are important in computer graphics. See [23, 24].

EXAMPLE 2.2.3 Our last example of a geometrical linear transformation

arises from projecting the plane onto a line l through the origin, inclined

at angle µ to the positive x{axis. Again we reduce that problem to the

simpler case where l is the x{axis and the equations of transformation are

x1 = x; y1 = 0.

In terms of matrices, we get transformation equations

·

x1

y1

¸

=

·

cos µ ¡sin µ

sin µ cos µ

¸ ·

1 0

0 0

¸ ·

cos (¡µ) ¡sin (¡µ)

sin (¡µ) cos (¡µ)

¸ ·

x

y

¸

2.3. RECURRENCE RELATIONS 31

=

·

cos µ 0

sin µ 0

¸ ·

cos µ sin µ

¡sin µ cos µ

¸ ·

x

y

¸

=

·

cos2 µ cos µ sin µ

sin µ cos µ sin2 µ

¸ ·

x

y

¸

:

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