Subject Expert: Bijumon R

7.5.2012

Subject Expert: Bijumon R

Content Editor: Nandakumar

PRESENTER:RASHMI.M

Abstract Algebra 7

Inner Product Spaces

Objectives

From this unit a learner is expected to achieve the following

1. Learn the concept of inner product and inner product spaces.

2. Familiarize with some inner products and inner product spaces.

3. Learn the definition of the Norm of a Vector Associated with an inner product.

4. Familiarize with some fundamental results, such as Polarization identity and Cauchy-Schwarz Inequality.

Sections

1. Introduction

2. Inner Proudct

3. Norm of a Vector Associated with an inner product

4. Matrix of the Inner Product in an ordered basis

1. Introduction

We know that in general a vector space is defined over an arbitrary field F. In this session we restrict to be the field of real or complex numbers. In the first case, the vector space is called real vector space and in the second case it is called a complex vector space. We are familiar with real vector spaces in analytical geometry and vector analysis. There we discuss the concept of length and orthogonality (perpendicularity). In this session our object is to study vector spaces in which it makes sense to speak of the ‘length’ of a vector. We shall do this by studying a certain type of scalar-valued function on a pair of vectors, known as an inner product that associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors.

Definition (inner product) Let be the field of real numbers or the field of complex numbers, and a vector space over . An inner product on is a function

which satisfies for and for the following:

(i)

(ii)

(iii) , where the bar denotes the complex conjugation

(iv) if .

Definition (inner product space) An inner product space is a real or complex vector space, together with a specified inner product on that space.

The following result is an immediate consequence of the Definition of inner proudct.

Theorem 1 If is an inner product on a vector space, then

for and .

Proof

, by (iii) in the definition of inner product

, by (i) in the definition of inner product

, by (ii) in the definition of inner product

, by the property of conjugation

, again by (iii)

Remarks

· When the field is , the complex conjugates appearing in (iii) in the definition and in the theorem are superfluous.

· When the field is , the complex conjugates appearing in (iii) in the definition and in the theorem are necessary for the consistency of the conditions because without these complex conjugates, we would have the following contradiction:

and .

· If 0 is the zero vector, then , where the 0 on the RHS is the scalar 0.

Notation In the examples that follow the field is either or .

We recall that when n be a positive integer, n , the set of all ordered n-tuples (x1, x2, … , xn) where xi Î , is a vector space over , where the vector addition and scalar multiplication are defined as below:

For = (x1, x2, …, xn) and = (y1, y2, …, yn) Î n and c Î

addition in n is defined by

+ = (x1 + y1, x2 + y2, …, xn + yn)

and the scalar multiplication in n is defined by

c· = (cx1, cx2, …, cxn ).

In the first example of this session we show that n is an inner product space.

Example 1 On , define by

, . . . (1)

where and .Verify that is an inner product and hence is an inner product space.

Solution

Let ,

(i)

(ii)

(iii)

, where the bar denotes the complex conjugation

(iv) If , then at least one and for it and so .

Now

. . . (2)

Being the sum of squares of non-negative values, the right hand side of (2) is greater than or equal to 0. Since at least one , the sum cannot be zero. So the sum is always greater than 0.

i.e., if .

That is we have verified that is an inner product. This inner product is called the standard inner product. Together with this inner product becomes an inner product space.

Remarks

· The inner product defined in (1) above can also be written as

, . . . (3¢)

· When , (1) can also be written as

, . . . (4²)

or simply as

, . . . (5¢²)

· In the real case the standard inner product is often called the scalar product or dot product and is denoted by .

In a vector space, there may be more than one inner product. In view of Example 1, on the vector space we have the standard inner product defined by

where and . In the following example we discuss another inner product on .

Example 2 For and , define

Verify that is an inner product.

Solution

The verification of conditions (i) to (iii) are left as an exercise to the student and we verify the condition (iv) only.

. . . (6)

(3) given above, being the sum of squares is always greater than or equal to 0. Hence if , then at least one is not equal to 0 and hence the above sum of squares is strictly greater than 0. i.e., if .

Hence is an inner product.

Example 3 Let , the vector space of all matrices over the field . Then we recall that V is isomorphic (in a natural way) to , the vector space of all tuples with coordinates from the field . Hence Example 1 suggests that for the equation

defines an inner product on V .

Recall that the conjugate transpose of a square matrix is the matrix defined by

i.e., th entry of is the complex conjugate of th entry of B.

Also, recall that trace of a square matrix is the sum of its diagonal entries.

Hence, the above inner product can be written as

because

Example 4 Let , the vector space of all (column) matrices over the field and let Q be an invertible matrix over . For every , define

On the right hand side of the above, we actually have a matrix and we identify it with its single entry.

Special case: When Q is the identity matrix, then the above reduces to

and this is the same inner product defined in Example 1 as can be seen below.

This inner product is called the standard inner product on .

Example 5 Let . i.e., V is the space of all continuous complex valued functions on the unit interval . For , define

Verify that is an inner product.

Solution

We only prove (iii) and (iv) and leaving the rest to the assignments.

(iii)

(iv)

If , then for some t in the interval and since is a continuous function, there exists an interval, say, I containing t such that for every x in the interval I. So for every x in that interval I of t and hence,

Hence

i.e., if , .

Theorem 2 Let be an inner product on a vector space V.

(a) (Polarization identity) For all

(b) Let . Then for all if and only if .

Proof:

(a) Since is linear in the first variable and conjugate-linear in the second, it is easy to check that the right side of the desired identity reduces to .

(b) If , then , so that for all Conversely, let for all Then, in particular, we have . By the positive-definiteness of , we see that .

2. Norm of a Vector

We now describe that an inner product on a real or complex vector space is determined by another function, known as quadratic form determined by the inner product. To define it, we first denote the positive square root of by Definition follows:

Definition (The norm a vector associated with the inner product) For any the norm of with respect to the inner product, denoted by is the positive square root of .

The standard inner product spaces in and suggests that it is appropriate to think of the norm of as the ‘length’ or ‘magnitude’ of .

Definition The quadratic norm determined by the inner product is the function that assigns to each vector the scalar

It follows from the properties of the inner product that

…(7)

and

…(8)

for all vectors and . In the real case by an algebraic manipulation, the above two equations gives the polarization identity

In the complex case we use the fact

and obtain from Equations (7) and (8), the polarization identity

3. Matrix of the Inner Product in an ordered basis

Suppose V be a finite dimensional vector space equipped with an inner product on it.

Let be an ordered basis for V, and that are given a particular inner product on V.

Claim: The inner product is completely determined by the values

If and then

where X, Y are the coordinate matrices of in the ordered basis , and G is the matrix with entries We call the matrix of the inner product in the ordered basis

We note that

so that G is Hermitian.

Theorem 3 If V is an inner product space, then for any vectors in V and any scalar , the following hold:

(i)

(ii)

(iii) (Cauchy-Schwarz inequality)

(iv)

Proof

(i) , by the definition of

, by the definition of the inner product

, by the definition of

(ii) , by the definition of

If , then by the property of inner product and hence by the above

(iii)

Case 1) When, and and so

Case 2) When, put

Then

= 0.

Also,

Hence

(iv)

, since

\ .

The function is the norm associated with the inner product on a vector space V if

Theorem 4 Equality in the Cauchy-Schwarz inequality ( (iii) in the above theorem ) holds if and only if and are linearly dependent.

Proof

Suppose equality in the Cauchy-Schwarz inequality holds; then going back ward in the proof of (iii), we have

implies since for an inner product

showing that is a scalar multiple of ; in other words and are linearly dependent.

Conversely, suppose that and are linearly dependent; then if = 0 then = 0 also and there is nothing to prove. Suppose

Then

. . . (9)

. . . (10)

From (6) and (7), we get , the equality in Cauchy-Schwarz inequality.

Remark In view of the theorem, unless

Example 6 If we apply Cauchy-Schwarz inequality to the inner products given in Examples 1, 2, 3 and 5 discussed above, we obtain the following inequalities:

3) .

Summary

In this session we introduced inner product spaces and its various properties with various examples. Consequently norm with respect to an inner product space and Matrix of the Inner Product in an ordered basis are discussed which are important tools in the coming sessions.before moving to the next session let us try to answer these questions .hope u enjoyed this class c u next time till then good bye

Assignments

1. Let V be an inner product space and let . Prove that if

2. Let be an inner product space. The distance between two vectors and in is defined by Show that

(a) (b) if and only if ;

3. Let be an inner product space, and let be vectors in . Show that if and only if for every in .

4. Find the inner product on such that .

5. If V be as in Example 5, using Cauchy-Schwarz inequality, prove that

Quiz

1. With respect to the standard inner product in , the value of is

(a) 2.

(b) 1.

(d) 5.

Ans. (c)

2. With respect to the standard inner product in , the value of is

(a) 2.

(b) 100.

(d) 400.

Ans. (c)

3. If is equipped with the innerproduct , then the value of is

(a) 1.

(b) 2.

(d) 4.

Ans. (b)

4. If is equipped with the innerproduct , then the angle between the vectors and is

(a) .

(b) .

(d) 4.

Ans. (b)

FAQ

1. The standard inner product in is the dot product in seen in geometry. Justify.

Ans. For the two vectors and in , the standard inner product is defined by

This is the usual dot product in seen in geometry. This can be seen as follows:

The dot product of two vectors and is given by Now the inner product and is also given by

(Ref. Fig.)

2. Give a non-Euclidean inner product on .

Ans. For the two vectors and in , define

The above defines a non-Euclidean inner product on .

3. Inner product is a bilinear operator when the field under consideration is . Justify.

Ans. Using (i) and (ii) of the Definition of an inner proudct, for and for

Hence inner product is linear in the first variable. Also, by Theorem 1,

showing that inner product is conjugate-linear in the second variable. If the field under consideration is then inner product is linear in the second variable. Since the inner product is linear in both of its arguments for real scalars, it may be called abilinear operatorin that context.

4. There are infinitely many inner products on Justify.

Ans. For any diagonal matrix with all for

defines an inner product on Hence, there are infinitely many different inner products on