Question

use the subspace theorem ( i) is it a non-empty space? ii) is it closed under vector addition? iii)is it closed under scalar multiplication?) to decide whether the following is a real vector space with its usual operations:

the set of all real polonomials of degree exactly n.

Answer 1

Let the set of all real polynomials of degree exactly n be denoted by P. Also, let p(x)=a_n xⁿ +a_n-1x^n-1+…+a₁x +a₀ and q(x) = b_nxⁿ +b_n-1x^n-1+…+b₁x+b₀ , where a_n and b_n are non-zero, be 2 arbitrary polynomials in P and let k be an arbitrary real scalar.

(i). P is apparently a non-empty as p(x) and q(x) belong to P.

(ii). We have p(x)+q(x) = a_nxⁿ +a_n-1x^n-1+…+a₁x +a₀ + b_nxⁿ +b_n-1x^n-1+…+b₁x +b₀ = (a_n+b_n)xⁿ+(a_n-1+b_n-1)x^n-1 +…+(a₁+b₁)x+( a₀+ b₀) which implies that p(x)+q(x) belongs to P so that P is closed under vector addition.

(ii). We have kp(x) = k(a_nxⁿ +a_n-1x^n-1+…+a₁x +a₀) = ka_nxⁿ +ka_n-1x^n-1+…+ka₁x +ka₀ which implies that kp(x) belongs to P so that P is closed under scalar multiplication.

(iii). The zero polynomial is of degree 0 so that it does not belong to P. Hence P is not a vector space.