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6.4 The Vector Space R n 163

where
O = (0,0,··· ,0)
n
is the zero vector in R (all components zero).
4. (α + β)F = αF + βF.
5. (αβ)F = α(βF).
6. α(F + G) = αF + αG.
7. αO = O.
n
Because of properties (1) through (7), and the fact that 1F = F for every F in R , we refer to
n
R as a vector space. There is a general theory of vector spaces which includes a broader class of
n
spaces than R . As one example, we will touch upon the function space C[a,b] in Section 6.5.
The norm (length) of an n-vector F =< x 1 , x 2 ,··· , x n > is

2
2
F = x + ··· + x .
1 n
n
This norm can be used to define a concept of distance in R . Given two points P :(x 1 , x 2 ,··· , x n )
n
and Q : (y 1 , y 2 ,··· , y n ) in R , think of
F =< x 1 , x 2 ,··· , x n > and G =< y 1 , y 2 ,··· , y n >
as vectors from the origin to these points, respectively. The distance between the points is the
norm of the difference of F and G:
distance between P and Q
= F − G

2
2
2
= (x 1 − y 1 ) + (x 2 − y 2 ) + ··· + (x n − y n ) .
3
When n = 3 this is the usual distance between two points in R .
The dot product of two n-vectors is defined by
< x 1 , x 2 ,··· , x n > · < y 1 , y 2 ,··· , y n >= x 1 y 1 + x 2 y 2 + ··· + x n y n .

3
This operation is a direct generalization of the dot product in R . Some properties of the norm
and the dot product are:
1. αF = |α| F .
2. Triangle inequality for n-vectors:
F + G ≤ F+ G .
3. F · G = G · F.
4. (F + G) · H = F · H + GH.
5. α(F · G) = (αF) · G = F · (αG).
2
6. F · F = F .
7. F · F = 0 if and only if F = O.
2
2
2
2
2
8. αF + βG = α F +2αβF · G + β G .
9. Cauchy-Schwarz inequality:
|F · G|≤ F G .

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October 14, 2010 14:21 THM/NEIL Page-163 27410_06_ch06_p145-186