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2.8 Composition and Invertibility of Linear Transformations The standard matrix of a linear transformation T can be used to find a generating set for the range of T.

Proof Property

Therangeofal ineartransf orm ation equalsthespanofthe columnsofits standardmatrix.

Range images

fS 1 S 2

Definition

Range fS 1 S 2 not onto onto

Definition(Review) Theorem 1.6Proof Theorem 2.10

Example: Is T onto? ! with the reduced row echelon form The system of linear equations Ax = b may be written as T

A (x) = b. Ax = b has a solution if and only if b is in the range of T A . T A is not onto if and only if " b such that Ax = b is inconsistent. T:R 3 R 3 withT x 1 x 2 x 3 x 1 +2x 2 +4x 3 x 1 +3x 2 +6x 3 2x 1 +5x 2 +10x 3

Definition(Review) Theorem 1.6Proof

one-to-one not one-to-one An equivalent condition for a function f to be one-to-one is that f (u) = f (v) implies u = v.

0 • • 0 0 • • 0 z •

Definition

one-to-one not one-to-one

0 • • 0 0 • • 0 z •

DefinitionDefinitionProperty

Example: Find a generating set for the null space of T. is the standard matrix of T, and has the reduced row echelon form The reduced row echelon form corresponds to the linear equations so a generating set for the null space of T is { [ 1 1 0 ]

T } the null space of T is the set of solutions to Ax = 0, where T:R 3 R 2 withT x 1 x 2 x 3 x 1 "x 2 +2x 3 "x 1 +x 2 "3x 3 x 1 !x 2 =0 x 3 =0

Proof Theorem 2.11(Review) Theorem 1.8

Example: The standard matrix A has rank 2 (nullity 1), so T is not one-to-one. If T A

is one-to-one and the solution of Ax = b exists, then the solution is unique. Conversely, if there is at most one solution to Ax = b for every b, then T

A is one-to-one. Example: row reduced echelon form rank A = 3 and nullity A = 2. ! T A is not onto and not one-to-one. ! Ax = b is inconsistent for b not in the range of T A , and the solutions of Ax = b is never unique. T:R 3 R 3 withT x 1 x 2 x 3 x 1 +2x 2 +4x 3 x 1 +3x 2 +6x 3 2x 1 +5x 2 +10x 3 =Ax A= 00133
23152
46162
46171
R=

11.5010

00130
00001 00000

Definition.Example: !

f x 1 x 2 x 2 1 x 1 x 2 x 1 +x 2 g x 1 x 2 x 3 x 1 !x 3 3x 2 (g!f) x 1 x 2 x 2 1 "(x 1 +x 2 3x 1 x 2 fgS 1 S 2 S 3 uf(u)g(f(u))···g!f

Example: ! T

B : reflection about the x-axis in R 2 . ! T A : rotation by 180 in R 2 . ! T BA : reflection about the y-axis in R 2 = rotation by 180 followed by reflection about the x-axis in R 2 . Proof A A B

Theorem 2.12

A= !10 0!1 B= 10 0!1 T BA u 1 u 2 =(BA) u 1 u 2 !10 01 u 1 u 2 !u 1 u 2

Definition.If A # R

n$n is invertible, then for all v # R n we have T A (T A -1(v)) = (T A T A -1) (v) = T AA -1(v) = T I n (v) = I n v = v, and T A -1(T A (v)) = v. Thus T A -1 = T A -1 .

Example:

A= 12 35
A !1 !52 3!1 T !1 A v 1 v 2 =T A !1 v 1 v 2 !52 3!1 v 1 v 2 !5v 1 +2v 2 3v 1 !v 2 T A v 1quotesdbs_dbs20.pdfusesText_26