Page 271 - A Course in Linear Algebra with Applications

7.4: The Method of Least Squares 255

Here it is assumed that we are working in the inner prod-
uct space C[—1, 1] where < , g > = J_ x f(x)g(x)dx. Let S
/
denote the subspace consisting of all quadratic polynomials in
x. An orthonormal basis for S was found in Example 7.3.6:

1 , [3 3>/5, 2 1-

x
By 7.4.7 the least squares approximation to e in S is simply
x
the projection of e on S; this is given by the formula
x
x
x
p = < e , h > h + < e , h > h + < e , h > h-
Evaluating the integrals by integration by parts, we obtain

x
<e',/i> = ^ J. \ <e J 2>=V6e- 1

and
x
1
<e ,h> = J\{e-le- ).
The desired approximation to e x is therefore

1 2
P=\(e-e-')+3e- x + ^(e-7e-')(x -±).

Alternatively one can calculate the projection by using the
2
standard basis 1, x,x .

Exercises 7.4
1. Find least squares solutions of the following linear systems:

xi + x 2 = 0

2 + X3 =
(a) I °° °
* xi - x 2 - x 3 = 3
£i + ^3 = 0