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Section 4.2
Linear Transformations
from 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 R
A 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)
f ( x1 , x2 ,, xn )  x  x    x
(A function from Rn to R)
2
1
2
2
2
n
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.
TRANSFORMATIONS
Let f1, f2, . . . , fm be real-valued functions of n
variables, say
w1  f1 ( x1 , x2 ,, xn )
w2  f 2 ( x1 , x2 ,, xn )

wm  f m ( x1 , x2 ,, xn )
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
T : R n  R m and
T ( x1 , x2 ,, xn )  ( w1 , w2 ,, wm )
If the equations are linear, the transformation
T: Rn → Rm is called a linear transformation
(or linear operator if m = n).
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,
 w1   a11 a12  a1n   x1 
 w  a
 x 
a

a
22
2n   2 
 2    21

 
    
  
 
w
a
a

a
mn   xn 
 m   m1 m 2
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
(TB  TA )( x)  TB (TA (x ))
 B( Ax)
 ( BA)x
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.