1. Section 4.2 Linear Transformations from R n to R m

2. 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.

3. FUNCTIONS FROM R n TO R A function from R n to R is a function that has n independent variables and gives only one output. Examples: f (x, y) = x 2 + xy + y 2 (A function from R 2 to R) (A function from R n to R)

4. FUNCTIONS FROM R n TO R m If the domain of f is R n and the range is in R m, then f is called a map or transformation from R n to R m, and we say the function maps R n to R m. We denote this by writing f : R n → R m NOTE: m can be equal to n in which case it function is called an operator on R n.

5. TRANSFORMATIONS Let f 1, f 2,..., f m be real-valued functions of n variables, say These equations assign a unique point (w 1, w 2,... w m ) in R m and define a transformation from R n to R m.

6. NOTATION AND LINEAR TRANSFORMATIONS If we denote the transformation by T, then If the equations are linear, the transformation T: R n → R m is called a linear transformation (or linear operator if m = n).

7. STANDARD MATRIX FOR A LINEAR TRANSFORMATION Let T: R n → R m and T(x 1, x 2,..., x n ) = (w 1, w 2,..., w m ) where w i = a i1 x 1 + a i2 x 2 +... + a in x n for 1 ≤ i ≤ m. In matrix notation, or w = Ax. The matrix A is called the standard matrix for the linear transformation T, and T is called multiplication by A.

8. SOME NOTATION If T: R n → R m 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 T A : R n → R m. Thus, T A (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 [T A ] = A

9. 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.

10. COMPOSITION OF LINEAR TRANSFORMATIONS If T A : R n → R k and T B : R k → R m are linear transformations, then the application of T A followed by T B produces a transformation from R n to R m. This transformation is called the composition of T B with T A, and is denoted by T B ◦ T A. Thus, (T B ◦ T A )(x) =T B (T A (x)).

11. LINEARITY OF T B ◦ T A The composition T B ◦ T A is linear since The above formula also tells us that the standard matrix for T B ◦ T A is BA. That is, T B ◦ T A = T BA.

12. COMPOSITIONS OF THREE OR MORE LINEAR TRANSFORMATIONS Compositions can be defined analogously for three or more linear transformations. (T 3 ◦ T 2 ◦ T 1 )(x) = T 3 (T 2 (T 1 (x))). Or, T C ◦ T B ◦ T A = T CBA.