Page 695 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

We have

and

where the numerical values have been rounded to one decimal place. Thus the estimated value of the spring constant is
pounds/inch.

Least Squares Fit of a Polynomial

The technique described for fitting a straight line to data points generalizes easily to fitting a polynomial of any specified
degree to data points. Let us attempt to fit a polynomial of fixed degree m

(8)

to n points

Substituting these n values of x and y into 8 yields the n equations

or, in matrix form, (9)
where

(10)

As before, the solutions of the normal equations

determine the coefficients of the polynomial. The vector v minimizes

Conditions that guarantee the invertibility of are discussed in the exercises. If is invertible, then the normal
(11)
equations have a unique solution , which is given by