Invertible Linear Mappings A mapping L : X Y is called invertible if - PowerPoint PPT Presentation

Invertible Linear Mappings

A mapping L : X → Y is called invertible if there exists L − 1 : Y → X such that L ◦ L − 1 = Id Y . L − 1 ◦ L = Id X , We call L − 1 the inverse of L . Theorem If L : X → Y is linear and invertible, then the inverse L − 1 is linear. 1

Theorem If L : X → Y is linear and invertible, then the inverse L − 1 is linear. Proof 1. Let α ∈ R and y ∈ Y . Set x := L − 1 ( y ). Since L ( α x ) = α y , we have α L − 1 ( y ) = α x = L − 1 ( α y ) . 2. Let y 1 , y 2 ∈ Y . Set x 1 = L − 1 ( y 1 ) and x 2 = L − 1 ( y 2 ). Since L ( x 1 + x 2 ) = y 1 + y 2 we have L − 1 ( y 1 + y 2 ) = x 1 + x 2 = L − 1 ( y 1 ) + L − 1 ( y 2 ) . Hence L − 1 is linear. 2

Theorem Let L : X → Y be an invertible linear mapping and let A ⊆ X. 1. A is linearly independent if and only if L ( A ) is linearly independent. 2. We have span L ( A ) = L (span A ) . 3. A is a basis if and only if L ( A ) is a basis. Corollary The image of a subspace under a linear mapping is again a subspace. Corollary Let S ⊆ X be a subspace of dimension k and let L : X → Y be an invertible linear mapping. Then L ( S ) is a subspace of dimension k. 3

Theorem Let v 1 , . . . , v n ∈ V and let L : V → W be invertible. Write w 1 = Lv 1 , w n = Lv n . . . . , Then v 1 , . . . , v n is linearly dependent if and only if w 1 , . . . , w n is linearly dependent . Proof. If α 1 , . . . , α n ∈ R with 0 = α 1 v 1 + · · · + α n v n , then 0 = L (0) = α 1 L ( v 1 ) + · · · + α n L ( v n ) = α 1 w 1 + · · · + α n w n . So linear dependence of v 1 , . . . , v n implies linear dependence of w 1 , . . . , w n . Since L is invertible and linear, the inverse L − 1 is linear. Hence the converse implication follows analogously. 4

Theorem Let v 1 , . . . , v n ∈ V and let L : V → W be invertible. Write w 1 = Lv 1 , w n = Lv n . . . . , Then v 1 , . . . , v n is a spanning set for V if and only if w 1 , . . . , w n is a spanning set for W . Proof. Let w ∈ W . There exists v ∈ V with L ( v ) = w . There exist α 1 , . . . , α n ∈ R with v = α 1 v 1 + · · · + α n v n . Consequently, w = L ( v ) = α 1 L ( v 1 ) + · · · + α n L ( v n ) = α 1 w 1 + · · · + α n w n . So v 1 , . . . , v n being a spanning set implies w 1 , . . . , w n being a spanning set. Since L is invertible and linear, the inverse L − 1 is linear. Hence the 5 converse implication follows analogously.

Basis Transformations

Let V be an n -dimensional vector space with two different bases: v 1 , . . . , v n , w 1 , . . . , w n . We let A = ( a ij ) ∈ R n × n be defined by v i = a i 1 w 1 + · · · + a in w n . If α 1 , . . . , α n , β 1 , . . . , β n ∈ R with α 1 v 1 + · · · + α n v n = β 1 w 1 + · · · + β n w n , then       a 11 . . . a n 1 α 1 β 1 . . . . ...  . .   .   .   = . . . .      a 1 n . . . a nn α n β n 6

      a 11 a n 1 . . . α 1 β 1 . . . . ...  . .   .   .   = . . . .      a 1 n . . . a nn α n β n Indeed, � n � n n n n � � � � � α i v i = α i a ij w j = a ij α i w j i =1 i =1 j =1 j =1 i =1 � �� � = β j 7

We call the matrix A the basis transition matrix from the basis v 1 , . . . , v n to the basis w 1 , . . . , w n . The basis transition matrix is necessarily invertible. Otherwise we had a linear dependence between basis vectors. The inverse of a basis transition matrix is again a basis transition matrix, with the roles of the bases reversed. 8

If we have three bases u 1 , . . . , u n , v 1 , . . . , v n , w 1 , . . . , w n , and let B uv , B vw ∈ R n × n denote the basis transition matrices from u 1 , . . . , u n to v 1 , . . . , v n and from w 1 , . . . , w n to w 1 , . . . , w n , respectively, then B uw = B vw ◦ B uv ∈ R n × n is the basis transition matrix from u 1 , . . . , u n to w 1 , . . . , w n . 9

Every invertible matrix can be thought of as basis transition matrix with respect to some bases: Pick A ∈ R n × n invertible and a basis v 1 , . . . , v n . Then w 1 = Av 1 , . . . , w n = Av n , is a basis, and A − 1 is the basis transition matrix from the basis v 1 , . . . , v n to the basis w 1 , . . . , w n . If w 1 , . . . , w n were linearly dependent with some coefficients β 1 , . . . , β n , then an application of the inverse matrix A − 1 would give coefficients α 1 , . . . , α n that yield a linear dependence of v 1 , . . . , v n . 10

Questions? 10