Let Px denote the vector space of all polynomials Let E be t

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