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Kernel and Range

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

If A is an m × n matrix and TA is the linear transformation defined by multiplication by A , the range of TA is the column space of A and the nullspace of A is the subspace { x ∈ R n^ | A x = 0 }. Today we extend these notions to general linear transformations between abstract vector spaces.

Examples

Example Suppose T : R^2 → R^2 is the projection on the x -axis operator. Find its kernel and range.

Example Suppose T : C(R) → C^1 (R) is given by T ( f ) =

∫ (^) x 0 f^ ( t )^ dt. Find its kernel and range.

Examples

Example Suppose T : R^2 → R^2 is the projection on the x -axis operator. Find its kernel and range.

Example Suppose T : C(R) → C^1 (R) is given by T ( f ) =

∫ (^) x 0 f^ ( t )^ dt. Find its kernel and range.

Examples (continued)

Example Suppose T : P 3 → M 22 is defined by

T ( a 0 + a 1 x + a 2 x^2 + a 3 x^3 ) =

[

a 0 + a 2 a 0 + a 3 a 1 + a 2 a 1 + a 3

]

Find its kernel and range.

Example Suppose T : P 2 → R is given by T ( p ) = p ( 2 ). Find its kernel and range.

Properties of Kernel and Range

Theorem If T : V → W is a linear transformation, then (^1) ker( T ) is a subspace of V. (^2) R ( T ) is a subspace of W.

Proof.

Rank and Nullity

Definition If T : V → W is a linear transformation, the dimension of R ( T ) is called the rank of T (denoted rank( T )). The dimension of the kernel of T is called the nullity of T (denoted nullity( T )).

Theorem If A is an m × n matrix and TA : R n^ → R m^ is multiplication by A, then (^1) nullity( TA ) = nullity( A ) , (^2) rank( TA ) = rank( A ).

Rank and Nullity

Definition If T : V → W is a linear transformation, the dimension of R ( T ) is called the rank of T (denoted rank( T )). The dimension of the kernel of T is called the nullity of T (denoted nullity( T )).

Theorem If A is an m × n matrix and TA : R n^ → R m^ is multiplication by A, then (^1) nullity( TA ) = nullity( A ) , (^2) rank( TA ) = rank( A ).

Examples

Example Find the rank and nullity for the linear transformation TA : R^3 → R^3 where

A =

Example Let D : P 3 → P 2 be given by D ( p ) = p ′( x ). Find the rank and nullity of D.

Dimension Theorem for Linear Transformations

Theorem (Dimension Theorem for Linear Transformations) If T : V → W is a linear transformation from an n-dimensional vector space V to a vector space W , then

rank( T ) + nullity( T ) = n.

Proof.

Homework

Read Section 8.2 and work exercises 1–9, 14, 17, 19, 22, 24, 26, 27.