Please help me write the process in detail, preferably by hand, thank you so much!

Part c: 2 -1 0 i: [2 points] Consider V = R3, and Let A = 1-1 2 -1 Define (u, v) = 0 2 u Av. Show that S equipped with this operation is an inner product space ii: [2 points] Let x E R and decide whether the following matrix is orthog- COST sin x onal: - sin x costPart a: Consider the vector space of real 3 x 3 matrices M3,3. i: [2 points] Consider S = {A ( M3,3/A = A"} Show that S is a subspace of V ii: [2 points] Find a basis for this subspace, and show it is in fact a basis Part b: Let V = Sege, and consider the subspace S = {a E Seqr| shift (shift (shift (shift(a)))) = a} i: [2 points] Show that S is a subspace of V. ii: [2 points] Find a basis for this subspace, and show it is in fact a basis iii: [4 points] For a = (do, a1, a2, a3, . . .; ), b = (bo, b1, b2, b3, . . . ), Define (a, b) = aobo + a1b1 + a2b2 + a363. Show that S equipped with this operation is an inner product space. iv: [1 point] Find an orthogonal basis for S with respect to this inner prod- uct