Page 261 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

Notes
n
Example 8: The standard basis of either R or C is an orthonormal set with respect to the
n
standard inner product.

2
Example 9: The vector (x, y) in R is orthogonal to (–y, x) with respect to standard inner
product, for
x
x
(( , )|( y , )) = xy yx 0.
y
2
However if R is equipped with the inner product of Example 2, then (x, y) and (– y, x) are
orthogonal if and only if
y = x

pq
Example 10: Let V be C n n , the space of complex n n matrices, and let E be the matrix
pq
whose only non-zero entry is a 1 in row p and column q. Then the set of all such matrices E is
orthonormal with respect to the inner product given in Example 3. For
pq
(E pq |E rs ) = tr(E E sr ) qs tr(E pr ) qs pr .

Example 11: Let V be the space of continuous complex-valued (or real-valued) functions
on the interval 0 x 1 with the inner product

1
(f|g) = f ( ) ( ) dx .
g
x
x
0
x
.
Suppose f n ( ) 2 cos 2 nx and that g n ( ) 2 sin 2 nx Then {1, f , g , f , g , ...} is an infinite
x
1
1
2
2
orthonormal set. In the complex case, we may also form the linear combinations
1
( f n ig n ), n 1, 2 ....
2
In this way we get a new orthonormal set S which consists of all functions of the form
x
h n ( ) e 2 inx , n 1, 2, ....
The set S’ obtained from S by adjoining the constant function 1 is also orthonormal. We assume
here that the reader is familiar with the calculation of the integrals in equation.
The orthonormal sets given in the examples above are all linearly independent. We show now
that this is necessarily the case.
Theorem 2: An orthogonal set of non-zero vectors in linearly independent.
Proof: Let S be a finite or infinite orthogonal set of non-zero vectors in a given inner product
space. Suppose . . . , are distinct vectors in S and that
1 2 m
c 1 1 c 2 2 ... c m m .
Then

( | ) = c |
k j j k
j

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