Let H={p(t) : p(t) =a+bt +ct^3; a,b,c ∈ R} (a) Show that H is a subspace of P 3 (b) Let p1. P2. P3 be polynomials in H such that p1(t)=2, p2(t) =(1+3t^3), p3(t)= -1-t-t"^3. Use coardinate vectors in...

Let H={p(t) : p(t) =a+bt +ct^3; a,b,c ∈ R}

(a) Show that H is a subspace of P 3

(b) Let p1. P2. P3 be polynomials in H such that p1(t)=2, p2(t) =(1+3t^3), p3(t)= -1-t-t"^3. Use coardinate vectors in order each of the following andjustify your answer each part.

(I) Verify that {p1, P2, P3} form a linearly independent set in P3.

(ii)  Verify that {p1, p2, P3} does not span P3.

(iii)  Can the set {p1, p2, P3} form basis for P3?