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182 CHAPTER 6 Vectors and Vector Spaces

for a ≤ x ≤ b. Linear independence means that the only way a linear combination of f 1 ,··· , f n
can be the zero function is for all the coefficients to be zero. This is the same as asserting that
no f j is a linear combination of the other functions. We saw this concept, without reference
to the vector space context, when dealing with solutions of second order linear homogeneous
differential equations in Chapter 2.
n
n
One significant difference between C[a,b] and R is that R has a basis consisting of n
vectors, hence has dimension n. However, C[a,b] has no such finite basis. Consider, for example,
the functions
3
n
2
p 0 (x) = 1, p 1 (x) = x, p 2 (x) = x , p 3 (x) = x ,··· , p n (x) = x ,
with n any positive integer. These functions are all in C[a,b] and are linearly independent. The
reason for this is that, if
n
2
c 1 + c 2 x + c 3 x + ··· + c n x = 0
for all x in [a,b], then each c i = 0 because a real polynomial of degree n can have at most n
distinct roots. We can produce arbitrarily large linearly independent sets of functions in C[a,b],
hence C[a,b] can have no finite basis.
We can introduce a dot product for functions in C[a,b] as follows. Select a function p that
is continuous on [a,b], with p(x)> 0for a < x < b.If f and g are in C[a,b], define
b
f · g = p(x) f (x)g(x)dx.
a
This operation is called a dot product with weight function p, and it has all of the properties we
saw for dot products of vectors. In particular:
1. f · g = g · f ,
2. ( f + g) · h = f · h + g · h,
3. c( f · g) = (cf ) · g = f · (cg),
4. f · f ≥ 0, and f · f = 0 if and only if f (x) = 0for a ≤ x ≤ b.
n
In view of property (4), we can, as in R , define the norm or length of f to be

b

2
f = f · f = p(x)( f (x)) dx.
a
Once we have the norm of a function, we can define the distance between f and g to be the norm
of f − g.Thisis

f − g = ( f − g) · ( f − g)

b

2
= p(x)( f (x) − g(x)) dx.
a
n
Continuing the analogy with R , define f and g to be orthogonal if f · g = 0. This means
that
b
p(x) f (x)g(x)dx = 0.
a
These definitions enable us to think geometrically in the function space C[a,b], with con-
cepts of distance between functions and orthogonality. The Gram-Schmidt process extends
verbatim to subspaces of C[a,b] using this integral dot product.

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October 14, 2010 14:21 THM/NEIL Page-182 27410_06_ch06_p145-186