Page 324 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
P. 324
4.4 In this section we shall apply our new knowledge of linear transformations to
polynomials. This is the beginning of a general strategy of using our ideas about
LINEAR
TRANSFORMATIONS to solve problems that are in different, yet somehow analogous, settings.
AND POLYNOMIALS
Polynomials and Vectors
Suppose that we have a polynomial function, say
where x is a real-valued variable. To form the related function we multiply each of its coefficients by 2:
That is, if the coefficients of the polynomial are a, b, c in descending order of the power of x with which they are associated,
then is also a polynomial, and its coefficients are , , in the same order.
Similarly, if is another polynomial function, then is also a polynomial, and its coefficients are
,, . We add polynomials by adding corresponding coefficients.
This suggests that associating a polynomial with the vector consisting of its coefficients may be useful.
EXAMPLE 1 Correspondence between Polynomials and Vectors
Consider the quadratic function . Define the vector
consisting of the coefficients of this polynomial in descending order of the corresponding power of x. Then multiplication of
by a scalar s gives , and this corresponds exactly to the scalar multiple
of z. Similarly, is , and this corresponds exactly to the vector sum :
In general, given a polynomial we associate with it the vector
