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Generalized Linear Transformations

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

We have already studied the properties of linear transformations whose domain and codomain are R m^ and R n. Today we will study linear transformations between abstract vector spaces.

Examples

Example If V and W are any two vector spaces the mapping T : V → W defined as T ( v ) = 0 for all v ∈ V is called the zero transformation.

Example If V is any vector space the mapping T : V → V defined as T ( v ) = v for all v ∈ V is called the identity operator.

Examples

Example If V and W are any two vector spaces the mapping T : V → W defined as T ( v ) = 0 for all v ∈ V is called the zero transformation.

Example If V is any vector space the mapping T : V → V defined as T ( v ) = v for all v ∈ V is called the identity operator.

Examples (continued)

Example If V is any vector space and k is a fixed scalar, the mapping T : V → V defined as T ( v ) = k v is a linear operator. If k > 1 this is called a dilation of V. If 0 < k < 1 this is called a contraction of V.

Example Suppose W is a finite dimensional subspace of an inner product space V. The mapping T : V → W defined as T ( v ) = proj W v is called the orthogonal projection of V onto W. If B = { w 1 , w 2 ,... , w n } is an orthonormal basis for W then

T ( v ) = proj W v = 〈 v , w 1 〉 w 1 + 〈 v , w 2 〉 w 2 + · · · 〈 v , w n 〉 w n.

Examples (continued)

Example If B = { w 1 , w 2 ,... , w n } is a basis for a finite dimensional vector space V , then the mapping T : V → R n^ defined by T ( v ) = ( v )B is a linear transformation.

Example The mapping T : Pn → Pn given by T ( p ) = T ( p ( x )) = p ( x − 1 ) is a linear operator.

Examples (continued)

Example The mapping D : C^1 (R) → C(R) defined by D ( f ) = f ′( x ) is a linear transformation.

Example

The mapping J : C(R) → C^1 (R) defined by J ( f ) =

∫ (^) x

0

f ( t ) dt is a linear transformation.

Examples (continued)

Example The mapping D : C^1 (R) → C(R) defined by D ( f ) = f ′( x ) is a linear transformation.

Example

The mapping J : C(R) → C^1 (R) defined by J ( f ) =

∫ (^) x

0

f ( t ) dt is a linear transformation.

Properties of Linear Transformations

Remark: in general linear transformations preserve linear combinations.

Theorem If T : V → W is a linear transformation, then: (^1) T ( 0 ) = 0 (^2) T (− v ) = − T ( v ) for all v ∈ V (^3) T ( u − v ) = T ( u ) − T ( v ) for all u , v ∈ V.

Proof.

Linear Transformations and Basis Vectors

Remark: a linear transformation is completely determined by what it does to the basis vectors of the domain space. Suppose T : V → W is a linear transformation where V is a finite dimensional vector space with basis vectors B = { v 1 , v 2 ,... , v n }. Suppose T ( v 1 ) = w 1 T ( v 2 ) = w 2 .. . T ( v n ) = w n then if v ∈ V there exist scalars c 1 , c 2 ,... , cn such that v = c 1 v 1 + c 2 v 2 + · · · + cn v n and then T ( v ) = T ( c 1 v 1 + c 2 v 2 + · · · + cn v n ) = c 1 T ( v 1 ) + c 2 T ( v 2 ) + · · · + cnT ( v n ) = c 1 w 1 + c 2 w 2 + · · · + cn w n

Composition of Linear Transformations

Definition If T 1 : U → V and T 2 : V → W are linear transformations then the composition of T 2 with T 1 denoted T 2 ◦ T 1 is the function

( T 2 ◦ T 1 )( u ) = T 2 ( T 1 ( u )),

where u ∈ U.

Theorem If T 1 : U → V and T 2 : V → W are linear transformations, then ( T 2 ◦ T 1 ) : U → W is also a linear transformation.

Proof.

Composition of Linear Transformations

Definition If T 1 : U → V and T 2 : V → W are linear transformations then the composition of T 2 with T 1 denoted T 2 ◦ T 1 is the function

( T 2 ◦ T 1 )( u ) = T 2 ( T 1 ( u )),

where u ∈ U.

Theorem If T 1 : U → V and T 2 : V → W are linear transformations, then ( T 2 ◦ T 1 ) : U → W is also a linear transformation.

Proof.

Example

Example Consider the linear transformations D : C^1 (R) → C(R) and J : C(R) → C^1 (R). Evaluate ( J ◦ D )( f ) where f = f ( x ) ∈ C^1 (R).

Homework

Read Section 8.1 and work exercises 1–9 odd, 12, 15, 19, 25, 29, 30, 31.