Consider rational functions of the form f ( x ) = p (

Kasen Keller

Kasen Keller

Answered question

2022-06-01

Consider rational functions of the form f ( x ) = p ( x ) q ( x ) where p is a polynomial of degree not exceeding n and q is a particular polynomial q ( x ) = ( x 1 ) ( x 2 ) ( x m ). Do these functions form a vector space? I think these function forms a vector space because they are closed under addition and scaling by a number. In case they do, what is the dimension of this vector space? Could you suggest a basis and write down an explicit formula for the coordinate functions?

Answer & Explanation

Pettanicej4lyy

Pettanicej4lyy

Beginner2022-06-02Added 4 answers

Let V := R deg n [ x ] be the vector space of polynomials with degree n.
Let W the set of polynomials you describe. It is clearly a vector space.
In short: "Whether you divide your polynomials by q ( x ) or not, is doesn't change anything".
More precisely,
(1) p ( x ) V         p ( x ) / q ( x ) W
is an isomorphism (immediate proof). In particular, V and W share the same dimension ( n + 1 ).
Isomorphism (1) provides as well a bases correspondence between the canonical basis { 1 , x , x 2 , x n } of V and:
{ 1 q ( x ) , x q ( x ) , x 2 q ( x ) , x n q ( x ) }
which is thus a basis for W ..

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