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(1) < u , v >=
(if k = R , then this translates to < u , v >=< v , u >, that is, <,> is
<>
vu
,
symmetric).
(2) < u , u >>0 for all nonzero vectors u .
An inner product space is a pair ( V ,<,>), where V is a vector space and <,> is an inner
product on V .
Note that < u , u > is always a real number by condition (1), so that (2) makes sense.
It is easy to show that a general inner product also satisfies
<>=
0u
,
0 for all
,
u
,
and
<>=
uu
,
0
if and only if
u
=
0
.
The (standard) dot product on R n is defined by
Definition.
uv
=
uv
+
u v
+◊◊◊+
u nn
.
11
2 2
The (standard) dot product on C n is defined by
Definition.
u 1 v 1 + u 2 v 2 + ···+ u n v n .
uv
=
The dot products on R n and C n are inner products.
C.2.1. Theorem.
Proof.
It is easy to check that the properties in the definition of a dot product are
satisfied.
If all one wants to do is to have a dot product in R n , we could have dispensed with
the definition of an inner product and simply shown that the dot product satisfies the
properties listed in the definition. However, the point to abstracting the basic prop-
erties into a definition is that it isolates the essential properties of an inner product
and one does not get sidetracked by details. Vector spaces admit many different func-
tions that satisfy the definition of an inner product.
An inner product on a vector space enables us to give a simple definition of the
length of a vector.
Definition. Let V be a vector space with an inner product <,> and let v ΠV . Define
the length | v | of v by
v
=<
v v
,
>
.
A vector of length 1 is called a unit vector.
If one writes out this definition of the length of a vector for the standard dot
product on R n in terms of its coordinates, one sees that it is just the usual Euclidean
length; however, that is not the really important point. It is property (2) in the defi-
nition of a dot product that guarantees that our definition of length is well defined. It
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