Ch7_1

4.10 Inner Product Spaces

In this section, we extend those concepts of Rn such as: dot product of two vectors, norm of a vector, angle betweenvectors, and distance between points, to general vector space.

This will enable us to talk about the magnitudes of functionsand orthogonal functions. This concepts are used to approximate functions by polynomials – a technique that is used to implement functions on calculators and computers.

We will no longer be restricted to Euclidean Geometry,we will be able to create our own geometries on Rn.

Ch7_2

The dot product was a key concept on Rn that led to definitions of norm,angle, and distance. Our approach will be to generalize the dot product of Rn to a general vector space with a mathematical structure called an innerproduct. This in turn will be used to define norm, angle, and distance fora general vector space.

Ch7_3

Definition

An inner product on a real space V is a function that associates a number, denoted 〈 u, v 〉 , with each pair of vectors u and v of V. This function has to satisfy the following conditions for vectors u, v, and w, and scalar c.

1. 〈 u, v 〉 = 〈 v, u 〉 (symmetry axiom)

2. 〈 u + v, w 〉 = 〈 u, w 〉 + 〈 v, w 〉 (additive axiom)

3. 〈 cu, v 〉 = c 〈 u, v 〉 (homogeneity axiom)

4. 〈 u, u 〉 0, and 〈 u, u 〉 = 0 if and only if u = 0

(positive definite axiom)

A vector space V on which an inner product is defined is called aninner product space.Any function on a vector space that satisfies the axioms of an inner productdefines an inner product on the space. There can be many inner products on a given vector space.

Ch7_4

Example 1Let u = (x1, x2), v = (y1, y2), and w = (z1, z2) be arbitrary vectors in R2. Prove that 〈 u, v 〉 , defined as follows, is an inner product on R2.

〈 u, v 〉 = x1y1 + 4x2y2

Determine the inner product of the vectors (2, 5), (3, 1) under this inner product.Solution

Axiom 1: 〈 u, v 〉 = x1y1 + 4x2y2 = y1x1 + 4y2x2 = 〈 v, u 〉

Axiom 2: 〈 u + v, w 〉 = 〈 (x1, x2) + (y1, y2) , (z1, z2) 〉

= 〈 (x1 + y1, x2 + y2), (z1, z2) 〉

= (x1 + y1) z1 + 4(x2 + y2)z2

= x1z1 + 4x2z2 + y1 z1 + 4 y2z2

= 〈 (x1, x2), (z1, z2) 〉 + 〈 (y1, y2), (z1, z2) 〉

= 〈 u, w 〉 + 〈 v, w 〉

Ch7_5

Axiom 3: 〈 cu, v 〉 = 〈 c(x1, x2), (y1, y2) 〉 = 〈 (cx1, cx2), (y1, y2) 〉

= cx1y1 + 4cx2y2 = c(x1y1 + 4x2y2) = c 〈 u, v 〉

Axiom 4: 〈 u, u 〉 = 〈 (x1, x2), (x1, x2) 〉 = 04 22

21 xx

Further, if and only if x1 = 0 and x2 = 0. That is u = 0. Thus 〈 u, u 〉 0, and 〈 u, u 〉 = 0 if and only if u = 0.

The four inner product axioms are satisfied,

〈 u, v 〉 = x1y1 + 4x2y2 is an inner product on R2.

04 22

21 xx

The inner product of the vectors (2, 5), (3, 1) is

〈 (2, 5), (3, 1) 〉 = (2 3) + 4(5 1) = 14

Ch7_6

Example 2Consider the vector space M22 of 2 2 matrices. Let u and v defined as follows be arbitrary 2 2 matrices.

Prove that the following function is an inner product on M22.〈 u, v 〉 = ae + bf + cg + dh

Determine the inner product of the matrices .

hg

fe

dc

bavu ,

Solution

Axiom 1: 〈 u, v 〉 = ae + bf + cg + dh = ea + fb + gc + hd = 〈 v, u 〉Axiom 3: Let k be a scalar. Then

〈 ku, v 〉 = kae + kbf + kcg + kdh = k(ae + bf + cg + dh) = k 〈 u, v 〉 4)01()90()23()52( ,

09

25

10

32

09

25 and

10

32

Ch7_7

Example 3Consider the vector space Pn of polynomials of degree n. Let f and g be elements of Pn. Prove that the following function defines an inner product of Pn.

Determine the inner product of polynomials f(x) = x2 + 2x – 1 and g(x) = 4x + 1

1

0)()(g , dxxgxff

Solution

Axiom 1: fgdxxfxgdxxgxfgf ,)()()()( ,1

0

1

0

hghf

dxxhxgdxxhxf

dxxhxgxhxf

dxxhxgxfhgf

, ,

)()()]()([

)]()()()([

)()]()([ ,

1

0

1

0

1

0

1

0

Axiom 2:

Ch7_8

We now find the inner product of the functions f(x) = x2 + 2x – 1 and g(x) = 4x + 1

2

)1294(

)14)(12(14 ,12

1

0

23

1

0

22

dxxxx

dxxxxxxx

Ch7_9

Norm of a Vector

DefinitionLet V be an inner product space. The norm of a vector v is denoted ||v|| and it defined by

vv,v

The norm of a vector in Rn can be expressed in terms of the dot product as follows

) , , ,() , , ,()() , , ,(

2121

22121

nn

nn

xxxxxxxxxxx

Generalize this definition:The norms in general vector space do not necessary have geometric interpretations, but are often important in numerical work.

Ch7_10

Example 4Consider the vector space Pn of polynomials with inner product

The norm of the function f generated by this inner product is

Determine the norm of the function f(x) = 5x2 + 1.

1

0)()( , dxxgxfgf

1

0

2)]([, dxxffff

Solution Using the above definition of norm, we get

3

28

1

0

24

1

0

222

]11025[

]15[15

dxxx

dxxx

The norm of the function f(x) = 5x2 + 1 is .3

28

Ch7_11

Example 2’ Consider the vector space M22 of 2 2 matrices. Let u and v defined as follows be arbitrary 2 2 matrices.

It is known that the function 〈 u, v 〉 = ae + bf + cg + dh is an inner product on M22 by Example 2.

The norm of the matrix is

hg

fe

dc

bavu ,

2222, dcba uuu

Ch7_12

DefinitionLet V be an inner product space. The angle between two nonzero vectors u and v in V is given by

vu

vu,cos

The dot product in Rn was used to define angle between vectors. The angle between vectors u and v in Rn is defined by

vu

vucos

Angle between two vectors

Ch7_13

Example 5Consider the inner product space Pn of polynomials with inner product

The angle between two nonzero functions f and g is given by

Determine the cosine of the angle between the functionsf(x) = 5x2 and g(x) = 3x

1

0)()(, dxxgxfgf

gf

dxxgxf

gf

gf

)()(

,cos

1

0

Solution We first compute ||f || and ||g||.

3]3[3 and 5]5[51

0

21

0

222 dxxxdxxx

Thus

4

15

35

)3)(5(

)()(cos

1

0

21

0 dxxx

gf

dxxgxf

Ch7_14

Example 2” Consider the vector space M22 of 2 2 matrices. Let u and v defined as follows be arbitrary 2 2 matrices.

It is known that the function 〈 u, v 〉 = ae + bf + cg + dh is an inner product on M22 by Example 2.

The norm of the matrix is

The angle between u and v is

hg

fe

dc

bavu ,

2222, dcba uuu

22222222

,cos

hgfedcba

dhcgbfae

vu

vu

Ch7_15

Orthogonal VectorsDef. Let V be an inner product space. Two nonzero vectors u and v in V are said to be orthogonal if

0, vu

Example 6

Show that the functions f(x) = 3x – 2 and g(x) = x are orthogonal in Pn with inner product

.)()(,1

0 dxxgxfgf

Solution

0][))(23(,23 10

231

0 xxdxxxxx

Thus the functions f and g are orthogonal in this inner productSpace.

Ch7_16

Distance

DefinitionLet V be an inner product space with vector norm defined by

The distance between two vectors (points) u and v is defined d(u,v) and is defined by

vv,v

) ,( ),( vuvuvuvu d

As for norm, the concept of distance will not have direct geometrical interpretation. It is however, useful in numerical mathematics to be able to discuss how far apart various functions are.

Ch7_17

Example 7Consider the inner product space Pn of polynomials discussed earlier. Determine which of the functions g(x) = x2 – 3x + 5 or h(x) = x2 + 4 is closed to f(x) = x2.Solution

13)53(53 ,53,)],([1

0

22 dxxxxgfgfgfd

16)4(4 ,4,)],([1

0

22 dxhfhfhfd

Thus The distance between f and h is 4, as we might suspect, g is closer than h to f.

.4),( and 13),( hfdgfd

Ch7_18

Inner Product on Cn

For a complex vector space, the first axiom of inner product is modified to read . An inner product can then be used to define norm, orthogonality, and distance, as far a real vector space.Let u = (x1, …, xn) and v = (y1, …, yn) be element of Cn. The most useful inner product for Cn is

uvvu , ,

nn yxyx 11vu,

vuvu

u

vuvu

),(

0, if

11

d

xxxx nn

※

※

※

Ch7_19

Example 8Consider the vectors u = (2 + 3i, 1 + 5i), v = (1 + i, i) in C2. Compute(a) 〈 u, v 〉 , and show that u and v are orthogonal.(b) ||u|| and ||v||(c) d(u, v)

Solution

.orthogonal are and thus

055))(51()1)(32(, )(

vu

vu iiiiiia

3))(()1)(1(

392613)51)(51()32)(32( )(

iiii

iiiib

v

u

42375)61)(61()21)(21(

)61 ,21(

) ,1()51 ,32(),( )(

iiiu

ii

iiiidc vuvu

Ch7_20

Homework

Exercise 7.1:1, 4, 8(a), 9(a), 10, 12, 13, 15, 17(a), 19, 20(a)