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Section 4.2

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

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Section 4.2

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  1. Section 4.2 Linear Transformations from Rn to Rm

  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 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) (A function from Rn to R)

  4. 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 mapsRn 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.

  5. TRANSFORMATIONS Let f1, f2, . . . , fm be real-valued functions of n variables, say These equations assign a unique point (w1, w2, . . . wm) in Rm and define a transformation from Rn to Rm.

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

  7. 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, 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: 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

  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 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)).

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

  12. 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.

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