2. Let F be the vector space of continuous functions f : R. R and let H be the subspace of F spanned by {1, sin(t), cos(t), cos(20). (a) (Spt) Let K be the set of functions K = sin2(t) + boos2(t) : a, b E R}. Determine whether K is a vector subspace of H or not. (Hint: some standard trigonometric identities could be useful.) Justify your answer. (b) (Spt) Let L be the set of functions L = {sin(t) + a cos(2t) : a E R}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) (Spt) Consider R as a real vector space (under the usual real number addition and multiplica-tion). Let T : H —> R be the function given by T(f) = fir fit) di. Show that T is a linear transformation. (d) (Spt) Find a spanning set for ker(T). Justify your answer. (e) (Spt) Let D : H —> F be given by D(f) = , the first derivative. Show that D is a linear transformation. (f) (Spt) Find a spanning set for ker(D), where D is the linear transformation from part (e). Justify your answer.