Page 255 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

analysis deal with these spaces. In these concrete examples, we had the idea of length, secondly Notes
we had the idea of the angle between two vectors. These became special cases of the notion of a
3
dot product (often called a scalar or inner product.) of vectors in R . Given the vectors v = (x , x ,
1 2
3
x ) and w = (y , y , y ) in R the dot product of v and w is defined as
3 1 2 3
v.w = x y + x y + x y .
1 1 2 2 3 3
Note that the length of the vector v is given by v .v and the angle between v and w is given by
. v w
cos = .
. v v w .w
We list a few of the properties of a dot product:
1. . v v 0
2. . v w . w v

3. v .(aw bw ) av .w bv .w
for any vectors v, w and real numbers a, b. If now include the complex field we slightly modify
the above relations and list them as follows:

1. . v w . w v
2. . v v 0 and .v 0 if and only if v 0;
v
3. (au + bw).v = au.v + bw.v
4. ( u av bw ) a ( . ) b u .w
u
v
for all complex numbers a, b and all complex vectors u, v, w.
Definition. Let F be the field of real numbers or the field of complex numbers, and V a vector
over F. An inner product on V is a function which assigns to each ordered pair of vectors , in
V a scalar ( | ) in F in such a way that for all , , in V and all scalars C.

(a) ( | ) ( | ) ( | );

(b) (c | ) c ( | )

(c) ( | ) ( | ), the bar denoting complex conjugation;

(d) ( | ) 0 if 0.
It should be observed that conditions (a), (b) and (c) imply that

(e) ( |c ) c ( | ) ( | ).
In the examples that follow and throughout the unit F is either the field of real numbers or the
field of complex numbers.

(n)
Example 1: In F define, for = (x , x , ... x ) and =(y y ... y ), ( | ) x ,y + x y ...
1 2 n 1 2 n 1 2 2
x y n we call ( | ) the Standard Inner Product.
n
(2)
Example 2: In F define for = (x , x ) and = (y , y ),
1 2 1 2
( | ) = 2 x y x y x y x y .
,
1 1 1 2 2 1 2 2

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