In each of the following decide whether the given subset H o

In each of the following, decide whether the given subset H of the given vector space V is a subspace of V. Justify your answers carefully. V = P_2, H ={p(t)| the coefficient of t^2 is positive}. V = P_3, H ={p(t)| the constant term of p(t) is 0}.

Solution

(a). Let p (t) = at² +bt +c where a is positive, be an arbitrary element of H and let be an arbitrary scalar. Then p(t) = at²+bt+c will not belong to H if is negative. Hence H is not closed under scalar multiplication and, therefore, H is not a subspace of V.

(b) Let p(t) = at³ +bt²+ct be an arbitrary elements of H and let be an arbitrary scalar. Then p(t) = a (t)³ +b(t)²+c t (at³ +bt²+ct) i.e. p(t) p(t). Hence H is not closed under scalar multiplication and, therefore, H is not a subspace of V.