section 5.2: inner product spaces determine whether a
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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 .
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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
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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 .
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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
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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 .
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