Section 4.2 Linear Transformations from R n to R m.
Transcript of Section 4.2 Linear Transformations from R n to R m.
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.