Section 4.2

Linear Transformationsfrom Rn to Rm

DOMAIN, CODOMAIN, AND RANGE OF A FUNCTION

Let f be a function from the set A into the set B.

• The set A is called the domain of f.

• The set B is called the codomain of f.

• The subset of B consisting of all possible values for f as a varies over A is called the range of f.

FUNCTIONS FROM Rn TO RA function from Rn to R is a function that has n independent variables and gives only one output.

Examples:

f (x, y) = x2 + xy + y2 (A function from R2 to R)

(A function from Rn to R)

222

2121 ),,,( nn xxxxxxf

FUNCTIONS FROM Rn TO Rm

If the domain of f is Rn and the range is in Rm, then f is called a map or transformation from Rn to Rm, and we say the function maps Rn to Rm. We denote this by writing

f : Rn → Rm

NOTE: m can be equal to n in which case it function is called an operator on Rn.

TRANSFORMATIONSLet f1, f2, . . . , fm be real-valued functions of n variables, say

),,,(

),,,(

),,,(

21

2122

2111

nmm

n

n

xxxfw

xxxfw

xxxfw

These equations assign a unique point (w1, w2, . . . wm) in Rm and define a transformation from Rn to Rm.

NOTATION AND LINEAR TRANSFORMATIONS

If we denote the transformation by T, then

If the equations are linear, the transformationT: Rn → Rm is called a linear transformation (or linear operator if m = n).

),,,(),,,(

and:

2121 mn

mn

wwwxxxT

RRT

STANDARD MATRIX FOR A LINEAR TRANSFORMATION

Let T: Rn → Rm and T(x1, x2, . . . , xn) = (w1, w2, . . . , wm) where wi = ai1x1 + ai2x2 + . . . + ainxn for 1 ≤ i ≤ m.

In matrix notation,

nmnmm

n

n

m x

x

x

aaa

aaa

aaa

w

w

w

2

1

21

22221

11211

2

1

or w = Ax.

The matrix A is called the standard matrix for the linear transformation T, and T is called multiplication by A.

SOME NOTATION• If T: Rn → Rm is multiplication by A, and if it is important to

emphasize that A is the standard matrix for T, we shall denote the linear transformation by TA: Rn → Rm. Thus,

TA(x) = Ax

• Sometimes it is awkward to introduce a new letter for the standard matrix of a linear transformation. In such cases we will denote the standard matrix for T by the symbol [T]. Thus, we can write

T(x) = [T]x

• Occasionally, the two notations will be mixed, and we will write

[TA] = A

GEOMETRY OF LINEAR TRANSFORMATIONS

The geometry of linear transformation is given in the Tables 4.2.2 through 4.2.9 on pages 185-190.

COMPOSITION OF LINEAR TRANSFORMATIONS

If TA: Rn → Rk and TB: Rk → Rm are linear transformations, then the application of TA followed by TB produces a transformation from Rn to Rm. This transformation is called the composition of TB with TA, and is denoted by TB ◦ TA. Thus,

(TB ◦ TA)(x) =TB(TA (x)).

LINEARITY OF TB ◦ TA

The composition TB ◦ TA is linear since

x

x

xx

)(

)(

))(())((

BA

AB

TTTT ABAB

The above formula also tells us that the standard matrix for TB ◦ TA is BA. That is,

TB ◦ TA = TBA.

COMPOSITIONS OF THREE OR MORE LINEAR TRANSFORMATIONS

Compositions can be defined analogously for three or more linear transformations.

(T3 ◦ T2 ◦ T1)(x) = T3(T2(T1(x))).

Or,

TC ◦ TB ◦ TA = TCBA.