Consider the following spaces P space of all polynomials PN

Consider the following spaces: P:= {space of all polynomials} P_N:= {space of all polynomials of degree lessthanorequalto N} (a) Determine the dimension of P_2. (b) More generally, determine the dimension of P_N. (c) Show that P is not a finite dimensional space?

Solution

(a)   A basis for P₂ is { 1, x, x² } so, the dimension of P₂ is 3

.(b) Similarly, a basis for P_N is { 1, x, x²,…,x^N } Then, the dimension of P_N is N+1.

(c) The set of polynomials with coefficients in F is a vector space over F, denoted F[x]. Vector addition and scalar multiplication are defined in the obvious manner. If the degree of the polynomials is unrestricted then the dimension of F[x] is infinite. Let us assume that this is not so and that B = { Q₁ , Q₂ , …Q_k } is a basis for P. Let N = max { deg(Q₁ ), defg(Q₂ ), …deg (Q_k ) }. Then the highest degree of a polynomial is N and x^N+1 cannot be expressed as a linear combination of Q₁ , Q₂ , …,Q_k as none of Q₁ , Q₂ , …Q_k has a term of degree higher than x^N. Then, it follows that B is not a basis for P, which is a contradiction. Hence, If the degree of the polynomials is unrestricted then the dimension of F[x] = P is infinite