Let P[x] denote the vector space of all polynomials. Let E be the set of polynomials which only contain terms of even degree of even degree. Is E a subspace of P[x]? Explain. Solution Let E be the set as given in the problem E = { f(x)| f(x) contains only terms of even degree} = { g(x2)| g any polynomial in F[x]} If g and h are any polynomials in F[x], clearly g+h and cg are polynomials in F[x] and (g+h)(x2) and cg(x2) belong to E (for any scalar c in F). So E is closed under vector addition and scalar multiplication. So E is a subspace of F.