CHAPTER 4Inner Product & Orthogonality

CHAPTER OUTLINE

Introduction Norm of the Vector, Examples of Inner Product Space

- Euclidean n-space - Function Space , Polynomial Space

Angle between Vectors Orthogonal & Orthonormal Set Normalizing Vector Gram Schmidt Process

iu

nR baC , )(tP

INNER PRODUCT SPACE

Is a vector space, with an inner product

Satisfy the following 3 axioms, for all vectors

1) Conjugate symmetry:2) Linearity in the first argument:

3) Positive definiteness: with equality only for x=0.

FVV :,

FaVzyx scalars all,,,yxyx ,,

zyzxzyx

yxayax

,,,

,,

0, yx

V

NORM OF THE VECTOR ( ) iu

• Norm of the vector or length:

• The norm of the vector in ,denoted by ;

• Eg: If

21

21 ... xxui

2

1

v

vv 2R

v

21

21 vvv

5

2v

EUCLIDEAN N-SPACE( )

• Consider the vector space . The dot product or scalar product in is defined by:

• This function defines an inner product on .

• Eg: Find the Euclidean inner producta)

nRnR

nR

nn yxyxyxyx ..., 2211

nR

vu,)3,1,0( )2,8,4( vu

FUNCTION SPACE POLYNOMIAL SPACE

• The notation - used to denote the vector space of all continuous functions on the closed interval , where .

• Let are functions in , an inner product on :

• Eg: Consider in the polynomial space with inner product

baC ,

)(tP

baC ,

ba, bta )( and )( tgtf baC ,

baC ,

b

adttgtfgf )()(,

2)( and 53)( ttgttf )(tP

1

0)()(, dttgtfgf

a) Find

b) Find

gf ,

gf and

ANGLE BETWEEN VECTORS

For any nonzero vectors u and v in an inner product space, V, the angle between u and v is defined to be the angle θ such that and

Eg: Consider the vector

Find the angle θ between .

0

vu

vu,cos

3in )3,4,1( and )5,3,2( Rvu vu and

ORTHOGONAL SET Let V be an inner product space. The

vectors is said to be orthogonal if

Eg: Determine whether the given vectors are orthogonal

Vuu ji ,

jiuuuu jiji when 0,

)3,4,1( , )3,2,1( , )1,1,1( wvu

ORTHONORMAL SET The set is said to be orthonormal if it is

orthogonal and each of its vectors has norm 1, that is for all i.

Eg: Let

Determine whether S is an orthonormal set.

1iu

0 and 1... 221 jini uuxxu

3321 )1,0,0(),0,1,0(),0,0,1(,, RuuuS

NORMALIZING VECTOR If or equivalently ,

then u is called a unit vector and is said to be normalized.

To obtain a unit vector, every nonzero vector v in V, can be multiplied by the reciprocal of its length

Eg: Let

Normalize u and v.

1u 1, uu

vv

v1

4in )1,2,2,4(),2,4,3,1( Rvu

GRAM SCHMIDT PROCESS

GRAM SCHMIDT PROCESS

Eg: Use Gram Schmidt Process to find an orthonormal set from the set

3)4,2,1(),1,1,1(),0,1,1( RS