TOPIC :inner product spaces

BRANCH :civil-2

By Rajesh Goswami

Chapter 4Inner Product spaces

Chapter Outline

• Orthogonal & Orthonormal Set• Orthogonal basis• Gram Schmidt Process

Orthogonal Set

Let V be an inner product space. The vectors is said to be orthogonal if

Vuu ji ,

jiuuuu jiji when 0,

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.

1iu

0 and 1... 221 jini uuxxu

Orthonormal Bases: Gram-Schmidt Process • Orthogonal:

A set S of vectors in an inner product space V is called an orthogonal set if every pair of vectors in the set is orthogonal.

Orthonormal:

An orthogonal set in which each vector is a unit vector is called orthonormal.

jijiVS

ji

n

01

,

,,, 21

vv

vvv

0,

,,, 21

ji

n VSvv

vvv

ji

Note:

If S is a basis, then it is called an orthogonal basis or an orthonormal basis.

• Ex 1: (A nonstandard orthonormal basis for R3)

Show that the following set is an orthonormal basis.

31,

32,

32,

322,

62,

62,0,

21,

21

321

S

vvv

Sol:

Show that the three vectors are mutually orthogonal.

09

2292

92

0023

223

200

32

31

61

61

21

vv

vv

vv

Show that each vector is of length 1.

Thus S is an orthonormal set.

1||||

1||||

10||||

91

94

94

333

98

362

362

222

21

21

111

vvv

vvv

vvv

The standard basis is orthonormal.

Ex 2: (An orthonormal basis for )

In , with the inner product)(3 xP

221100, bababaqp

} , ,1{ 2xxB

)(3 xP

Sol:

,001 21 xx v ,00 2

2 xx v ,00 23 xx v

0)1)(0()0)(1()0)(0(, ,0)1)(0()0)(0()0)(1(, ,0)0)(0()1)(0()0)(1(,

32

31

21

vvvvvv

Then

1110000

,1001100

,1000011

333

222

111

v,vv

v,vv

v,vv

Gram Schmidt Process

• Gram-Schmidt orthonormalization process: is a basis for an inner product space V },,,{ 21 nB uuu

11Let uv })({1 1vw span

}),({2 21 vvw span

},,,{' 21 nB vvv

},,,{''2

2

n

nBvv

vv

vv

1

1

is an orthogonal basis.

is an orthonormal basis.

1

1 〉〈〉〈proj

1

n

ii

ii

innnnn n

vv,vv,vuuuv W

2

22

231

11

133333 〉〈

〉〈〉〈〉〈proj

2v

v,vv,uv

v,vv,uuuuv W

111

122222 〉〈

〉〈proj1

vv,vv,uuuuv W

Sol: )0,1,1(11 uv

)2,0,0()0,21,

21(

2/12/1)0,1,1(

21)2,1,0(

222

231

11

1333

vvvvuv

vvvuuv

Ex (Applying the Gram-Schmidt ortho normalization process)

Apply the Gram-Schmidt process to the following basis.

)}2,1,0(,)0,2,1(,)0,1,1{(321

Buuu

)0,21,

21()0,1,1(

23)0,2,1(1

11

1222

vvvvuuv

}2) 0, (0, 0), , 21 ,

21( 0), 1, (1,{},,{' 321

vvvB

Orthogonal basis

}1) 0, (0, 0), , 2

1 ,21( 0), ,

21 ,

21({},,{''

3

3

2

2

vv

vv

vv

1

1B

Orthonormal basis

Thus one basis for the solution space is

)}1,0,8,1(,)0,1,2,2{(},{ 21 uuB

1 ,2 ,4 ,3

0 1, 2, ,2 9181 0, 8, 1,

,,

0 1, 2, ,2

1

11

1222

11

vvvvuuv

uv

1,2,4,3 0,1,2,2' B (orthogonal basis)

301,

302,

304,

303 , 0,

31,

32,

32''B

(orthonormal basis)