Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER } (a) Show that H is a subspace of P3: (b) Let p 1, P2, P3 be polynomials in H, such that p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate vectors in...

Extracted text: Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER } (a) Show that H is a subspace of P3: (b) Let p 1, P2, P3 be polynomials in H, such that p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate vectors in 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 {p, P2, P3} does not span P3. (iii) Can the set {p,, pɔ, p3} form basis for P3? 11 (c) Let T:H→ R* be a linear transformation, defined by p(0) - p(0) T(p) = p(0) Lp(0) Find Ker T and find two polynomials that span Ker T.