Page 111 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
98
Inner Products
To motivate the concept of inner product, let’s think of vectors in R 2
3
and R as arrows with initial point at the origin. The length of a vec-
3
2
tor x in R or R is called the norm of x, denoted x . Thus for

2
2
2
If we think of vectors x = (x 1 ,x 2 ) ∈ R , we have x = x 1 + x 2 .
as points instead of
x -axis
arrows, then x 2
should be interpreted
as the distance from
the point x to the (x , x )
1 2
origin.
x
x -axis
1

2
2
The length of this vector x is x 1 + x 2 .

3 2 2 2
Similarly, for x = (x 1 ,x 2 ,x 3 ) ∈ R , we have x = x 1 + x 2 + x 3 .
Even though we cannot draw pictures in higher dimensions, the gener-
n
alization to R is obvious: we define the norm of x = (x 1 ,...,x n ) ∈ R n
by

2
2
x = x 1 +· · ·+ x n .
n
The norm is not linear on R . To inject linearity into the discussion,
n
we introduce the dot product. For x, y ∈ R , the dot product of x
and y, denoted x · y, is defined by
x · y = x 1 y 1 +· · ·+ x n y n ,
where x = (x 1 ,...,x n ) and y = (y 1 ,...,y n ). Note that the dot product
n 2
of two vectors in R is a number, not a vector. Obviously x · x = x
n
n
for all x ∈ R . In particular, x · x ≥ 0 for all x ∈ R , with equality if
n
and only if x = 0. Also, if y ∈ R is fixed, then clearly the map from R n
n
to R that sends x ∈ R to x · y is linear. Furthermore, x · y = y · x
n
for all x, y ∈ R .
An inner product is a generalization of the dot product. At this
point you should be tempted to guess that an inner product is defined