BRANCH :civil-2

TOPIC :INNER PRODUCT SPACES

Rajesh Goswami

CHAPTER 4Inner Product

spaces

Chapter Outline Introduction Norm of the Vector Examples of Inner Product Space Angle between Vectors Gram Schmidt Process

Inner Product Spaces Inner product:

Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms.

(1) (2) (3) (4)

〉〈〉〈 uvvu ,, 〉〈〉〈〉〈 wuvuwvu ,,,

〉〈〉〈 vuvu ,, cc 0, 〉〈 vv 0, 〉〈 vv

Note:

VRn

space for vectorproduct inner general, )for product inner Euclidean (productdot

vuvu

Note:A vector space V with an inner product is called an inner product space.

, ,V Vector space:Inner product space: , , , ,V

(Properties of inner products) Let u, v, and w be vectors in an inner product space V, and let c be any real number. (1) (2) (3)(4)(5)

0,, 〉〈〉〈 0vv0〉〈〉〈〉〈 wvwuwvu ,,,

〉〈〉〈 vuvu ,, cc

Norm (length) of u:

〉〈 uuu ,||||

〉〈 uuu ,|||| 2

Note:

, , ,, , ,u v w u w v wu v w u v u w

u and v are orthogonal if .

Distance between u and v:

vuvuvuvu ,||||),(d

Angle between two nonzero vectors u and v:

0,||||||||

,cosvuvu 〉〈

Orthogonal:

0, 〉〈 vu

)( vu

Distance between two vectors:The distance between two vectors u and v in

Rn is ||||),( vuvu d

Notes: (Properties of distance)(1)(2) if and only if (3)(4)

0),( vud0),( vud vu

),(),( uvvu dd

( , ) ( , ) (w, v)d u v d u w d

( , ) ( , ) (w, v)d u v d u w d

(Finding the distance between two vectors)The distance between u=(0, 2, 2) and v=(2, 0, 1) is

312)2(

||)12,02,20(||||||),(222

vuvud

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 0

vuvu,

cos

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

Eg: Consider the vector Find the angle θ between

vu and

(2,3,5) (1, 4,3)cos38

26

531 40

0.159

Orthogonal vectors:Two vectors u and v in Rn are orthogonal if 0vu

Note:The vector 0 is said to be orthogonal to every vector.

(Finding orthogonal vectors) Determine all vectors in Rn that are orthogonal to u=(4, 2).

024

),()2,4(

21

21

vv

vvvu

0

211024

tvtv

21 ,2

Rt,tt

,2

v

)2,4(u Let ),( 21 vvv Sol:

(The Pythagorean theorem)If u and v are vectors in Rn, then u and v are orthogonalif and only if

222 |||||||||||| vuvu

Dot product and matrix multiplication:

nu

uu

2

1

u

nv

vv

2

1

v

][][ 22112

1

21 nn

n

nT vuvuvu

v

vv

uuu

vuvu

(A vector in Rn is represented as an n×1 column matrix)

),,,( 21 nuuu u

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)