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5.4 Orthogonal Vectors 299

Another orthonormal basis B need not be directed in the same way as S, but
that’s the only significant difference because it’s geometrically evident that B
must amount to some rotation of S. Consequently, we should expect general
orthonormal bases to provide essentially the same advantages as the standard
n
basis. For example, an important function of the standard basis S for is to
provide coordinate representations by writing
 
x 1
 x 2 
x =[x] S =  .  to mean x = x 1 e 1 + x 2 e 2 + ··· + x n e n .
 . 
.
x n
With respect to a general basis B = {u 1 , u 2 ,..., u n } , the coordinates of x
are the scalars ξ i in the representation x = ξ 1 u 1 + ξ 2 u 2 + ··· + ξ n u n , and, as
illustrated in Example 4.7.2, finding the ξ i ’s requires solving an n × n system,
anuisance we would like to avoid. But if B is an orthonormal basis, then the
ξ i ’s are readily available because u i x = u i ξ 1 u 1 + ξ 2 u 2 + ··· + ξ n u n =
n 2 40
ξ
j=1 j u i u j = ξ i u i = ξ i . This yields the Fourier expansion of x.
Fourier Expansions
If B = {u 1 , u 2 ,..., u n } is an orthonormal basis for an inner-product
space V, then each x ∈V can be expressed as

x = u 1 x u 1 + u 2 x u 2 + ··· + u n x u n . (5.4.3)

This is called the Fourier expansion of x. The scalars ξ i = u i x
are the coordinates of x with respect to B, and they are called the
Fourier coefficients. Geometrically, the Fourier expansion resolves x
into n mutually orthogonal vectors u i x u i , each of which represents
the orthogonal projection of x onto the space (line) spanned by u i .
(More is said in Example 5.13.1 on p. 431 and Exercise 5.13.11.)

40
Jean Baptiste Joseph Fourier (1768–1830) was a French mathematician and physicist who,
while studying heat flow, developed expansions similar to (5.4.3). Fourier’s work dealt with
special infinite-dimensional inner-product spaces involving trigonometric functions as discussed
in Example 5.4.6. Although they were apparently used earlier by Daniel Bernoulli (1700–1782)
to solve problems concerned with vibrating strings, these orthogonal expansions became known
as Fourier series, and they are now a fundamental tool in applied mathematics. Born the son
of a tailor, Fourier was orphaned at the age of eight. Although he showed a great aptitude for
mathematics at an early age, he was denied his dream of entering the French artillery because
of his “low birth.” Instead, he trained for the priesthood, but he never took his vows. However,
his talents did not go unrecognized, and he later became a favorite of Napoleon. Fourier’s work
is now considered as marking an epoch in the history of both pure and applied mathematics.
The next time you are in Paris, check out Fourier’s plaque on the first level of the Eiffel Tower.