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Inverse Linear Transformations

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

We have already discussed the concept of invertibility in the context of linear transformations between R n^ and R n. Today we will focus on the invertibility of linear transformations between abstract vector spaces.

One-to-One

Definition A linear transformation T : V → W is one-to-one if T maps distinct vectors in V to distinct vectors in W.

Recall: if T is represented by multiplication by a square matrix, we already know that T is one-to-one if and only if the matrix is invertible.

Examples

Example Determine is each of the following linear transformations is one-to-one. (^1) D : C^1 (R) → C(R) where D ( f ) = f ′( x ). (^2) T : P 1 → R^2 where T ( p ) = ( p ( 0 ), p ( 1 )).

Equivalent Statements

Theorem If T : V → W is a linear transformation, then the following statements are equivalent. (^1) T is one-to-one. (^2) The kernel of T contains only the zero vector, i.e., ker( T ) = { 0 }. (^3) nullity( T ) = 0_._

Proof. (1) =⇒ (2)

Equivalent Statements

Theorem If T : V → W is a linear transformation, then the following statements are equivalent. (^1) T is one-to-one. (^2) The kernel of T contains only the zero vector, i.e., ker( T ) = { 0 }. (^3) nullity( T ) = 0_._

Proof. (1) =⇒ (2) =⇒ (1)

Finite-Dimensional Vector Spaces

If T : V → V and V is a finite-dimensional vector space, there is a fourth equivalent statement.

Theorem If T : V → V is a linear operator on a finite-dimensional vector space V , then the following statements are equivalent. (^1) T is one-to-one. (^2) The kernel of T contains only the zero vector, i.e., ker( T ) = { 0 }. (^3) nullity( T ) = 0_._ (^4) The range of T is V ; i.e., R ( T ) = V.

Proof. (3) =⇒ (4)

Finite-Dimensional Vector Spaces

If T : V → V and V is a finite-dimensional vector space, there is a fourth equivalent statement.

Theorem If T : V → V is a linear operator on a finite-dimensional vector space V , then the following statements are equivalent. (^1) T is one-to-one. (^2) The kernel of T contains only the zero vector, i.e., ker( T ) = { 0 }. (^3) nullity( T ) = 0_._ (^4) The range of T is V ; i.e., R ( T ) = V.

Proof. (3) =⇒ (4)

Examples

Example Use the previous theorem to determine if the following linear transformations are one-to-one. (^1) T : R^2 → R^2 given by

T ( x , y ) = ( x + y , x − y ).

(^2) T : R^2 → R^2 given by

T ( x , y ) = ( 2 x + y , x − 2 y ).

Inverse Linear Transformations

If T : V → W is a one-to-one linear transformation we can define the inverse of T , denoted T −^1 as a map T −^1 : R ( T ) → V where

T ( T −^1 ( w )) = w for all w ∈ R ( T ), and T −^1 ( T ( v )) = v for all v ∈ V.

Examples

Example Let T : P 3 → P 4 be given by T ( p ) = ( x + 1 ) p ( x ). Determine if T is one-to-one. If so, find T −^1.

Example Suppose Q is a fixed 2 × 2 invertible matrix. Let T : M 22 → M 22 be given by T ( A ) = Q −^1 AQ for all A ∈ M 22. Determine if T is one-to-one. If so, find T −^1.

Composition of Linear Transformations

Theorem If T 1 : U → V and T 2 : V → W are both one-to-one linear transformations, then (^1) T 2 ◦ T 1 is one-to-one. (^2) ( T 2 ◦ T 1 )−^1 = T (^) 1 − 1 ◦ T (^) 2 − 1_._

Proof.

In general

( T 1 ◦ T 2 ◦ · · · ◦ Tn )−^1 = T (^) n −^1 ◦ · · · ◦ T (^) 2 − 1 ◦ T (^) 1 − 1.

Composition of Linear Transformations

Theorem If T 1 : U → V and T 2 : V → W are both one-to-one linear transformations, then (^1) T 2 ◦ T 1 is one-to-one. (^2) ( T 2 ◦ T 1 )−^1 = T (^) 1 − 1 ◦ T (^) 2 − 1_._

Proof.

In general

( T 1 ◦ T 2 ◦ · · · ◦ Tn )−^1 = T (^) n −^1 ◦ · · · ◦ T (^) 2 − 1 ◦ T (^) 1 − 1.

Homework

Read Section 8.3 and work exercises 1, 3, 5, 7, 10, 11, 13, 18, 19, 20.