Page 325 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

in (Figure 4.4.1). It is then possible to view operations like as being equivalent to a linear transformation on
rather than on the polynomials themselves.
, namely . We can perform the desired operations in

Figure 4.4.1
The vector z is associated with the polynomial p.

EXAMPLE 2 Addition of Polynomials by Adding Vectors , we could define
Let and . Then to compute

and perform the corresponding operation on these vectors:

Hence .

This association between polynomials of degree n and vectors in would be useful for someone writing a computer program

to perform polynomial computations, as in a computer algebra system. The coefficients of polynomial functions could be stored as

vectors, and computations could be performed on these vectors.

For convenience, we define to be the set of all polynomials of degree at most n (including the zero polynomial, all the
coefficients of which are zero). This is also called the space of polynomials of degree at most n. The use of the word space
indicates that this set has some sort of structure to it. The structure of will be explored in Chapter 8.

EXAMPLE 3 Differentiation of Polynomials