CREATED BY SHANNON MARTIN GRACEY 134

Section 5.2: INNER PRODUCT SPACES

When you are done with your homework you should be able to…

Determine whether a function defines an inner product, and find the inner product of two vectors in nR , ,m nM , nP , and ,C a b

Find an orthogonal projection of a vector onto another vector in an inner product space

DEFINITION OF AN 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. , ___________________u v

2. , ___________________ u v w

3. , ___________________c u v

4. , 0, and , 0 iff _____________ v v v v

NOTE:

CREATED BY SHANNON MARTIN GRACEY 135

Example 1: Show that the function 1 1 2 2 3 3, 2u v u v u v u v defines an inner

product on 3R , where , 1 2 3, ,u u uu and 1 2 3, ,v v vv .

Example 2: Show that the function 1 1 2 2 3 3, u v u v u v u v does not define an

inner product on 3R , where , 1 2 3, ,u u uu and 1 2 3, ,v v vv .

CREATED BY SHANNON MARTIN GRACEY 136

THEOREM 5.7: 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. , _________ __________ 0 v

2. , ___________________ u v w

Proof:

3. , ___________________c u v

CREATED BY SHANNON MARTIN GRACEY 137

DEFINITION OF LENGTH, DISTANCE, AND ANGLE

Let u and v be vectors in an inner product space V .

1. The length (or ______________) of u is ________________________.

2. The distance between u and v is _______________________________.

3. The angle between and two vectors u and v is given by ________________________________________________________.

4. u and v are orthogonal when __________________________________.

If _________________, then u is called a _____________ vector. Moreover,

if v is any nonzero vector in an inner product space V , then the vector

__________________________ is a _______________ vector and is called

the __________________ vector in the ____________________ of v .

CREATED BY SHANNON MARTIN GRACEY 138

Example 3: Consider the following inner product defined on nR :

0, 6 u , 1,1 v , and 1 1 2 2, 2u v u v u v

a. Find ,u v

b. Find u

c. Find v

d. Find ,d u v

CREATED BY SHANNON MARTIN GRACEY 139

Example 4: Consider the following inner product defined:

1

1,f g f x g x dx

, f x x , 2 2g x x x

a. Find ,u v

b. Find f

c. Find g

d. Find ,d f g

CREATED BY SHANNON MARTIN GRACEY 140

THEOREM 5.8

Let u and v be vectors in an inner product space V .

1. Cauchy-Schwarz Inequality: _______________________________

2. Triangle Inequality: _____________________________________

3. Pythagorean Theorem: u and v are orthogonal if and only if ____________________________________________________

Example 5: Verify the triangle inequality for 0 12 1

A

, 1 12 2

B

, and

11 11 21 21 12 12 22 22,A B a b a b a b a b .

CREATED BY SHANNON MARTIN GRACEY 141

DEFINITION OF ORTHOGONAL PROJECTION

Let u and v be vectors in an inner product space V , such that v 0 . Then the orthogonal projection of u onto v is

THEOREM 5.9: ORTHOGONAL PROJECTION AND DISTANCE

Let u and v be vectors in an inner product space V , such that v 0 . Then

CREATED BY SHANNON MARTIN GRACEY 142

Example 6: Consider the vectors

1, 2 u and 4, 2v . Use the Euclidean inner product to find the following:

a. projvu

b. proju v

c. Sketch the graph of both projvu and proju v .