Page 327 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The corresponding operation from to takes to .

The relationship between an action on and its corresponding action on the vector of coefficients in , and the similarities
between and , will be explored in more detail later in this text.

Interpolating Polynomials

Consider the problem of interpolating a polynomial to a set of points , …, . That is, we seek to find a curve

of minimum degree that goes through each of these data points (Figure 4.4.2). Such a

curve must satisfy

Figure 4.4.2
Interpolation

Because the are known, this leads to the following matrix system:

Note that this is a square system when . Taking gives the following system for the coefficients of the interpolating
polynomial :

(1)

The matrix in 1 is known as a Vandermonde matrix; column j is the second column raised element wise to the power. The