. These coefficients, together with the basis functions, define the approximate solution un. The exact and approximate solutions are shown in Fig. 1.1.
The choice of basis functions
By definition, a set of functions
,
are linearly independent if
implies that
for
. It is left to the reader to show that if the basis functions are linearly independent then matrix
is invertible.
Given a set of linearly independent functions
,
, the set of functions that can be written as
is called the span and
are basis functions of S.
We could have defined other polynomial basis functions, for example;
(1.15)
When one set of basis functions
can be written in terms of another set
in the form:
(1.16)
where
is an invertible matrix of constant coefficients then both sets of basis functions are said to have the same span. The following exercise demonstrates that the approximate solution depends on the span, not on the choice of basis functions.
Exercise 1.1 Solve the problem of Example 1.1 using the basis functions
,
and show that the resulting approximate solution is identical to the approximate solution obtained in Example 1.1. The span of the basis functions in this exercise and in Example 1.1 is the same: It is the set of polynomials of degree less than or equal to 3 that vanish in the points
and
.
Summary of the main points
1 The definition of the integral by eq. (1.8) made it possible to find an approximation to the exact solution u of eq. (1.5) without knowing u.
l right-parenthesis period"/>
