Page 226 - A Course in Linear Algebra with Applications

2 1 0 Chapter Seven: Orthogonality in Vector Spaces

The understanding here is that these properties must hold for
all vectors u, v, w and all real scalars c, d.
We now give some examples of inner products, the first
one being the scalar product, which provided the original mo-
tivation.
Example 7.2.1
n
Define an inner product < > on R by the rule

T
< X, Y > = X Y.
That this is an inner product follows from the laws of matrix
T
algebra, and the fact that X X is non-negative and equals 0
only if X = 0. This inner product will be referred to as the
n
standard inner product on R . It should be borne in mind
that there are other possible inner products for this vector
space; for example, an inner product on R 3 is defined by

< X, Y >= 2xxv\ + 3x 2y 2 + 4z 3?/3

where X and Y are the vectors with entries x\, X2, x$ and y±,
J/2; 2/3 respectively. The reader should verify that the axioms
for an inner product hold in this case.

Example 7.2.2
Define an inner product < > on the vector space C[a,b] by
the rule

<f,9>= / f(x)g(x)dx.
J a
This is very different type of inner product, which is im-
portant in the theory of orthogonal functions. Well-known
properties of integrals show that the requirements for an in-
ner product are satisfied. For example,

rb
2
< / , / > = / f(x) dx>0
J a