For Problems 19-22, determine whether the given set of vectors is linearly independent in P2(R).

please do #21, 24, 25,28 thx with logic

24. Let S be the subspace of R3 that consists of all solu- tions to the equation x - 3y + z = 0. Determine a basis for S, and hence, find dim[S]. 25. Let S be the subspace of R3 consisting of all vectors of the form (r, r - 2s, 3s - 5r), where r and s are real numbers. Determine a basis for S, and hence, find dim[S]. 26. Determine a basis S for P3 (R), and hence, prove that dim[ P3 (R)] = 4. Be sure to prove that S is a basis. 27. Determine a basis S for P3 (R) whose elements all have the same degree. Be sure to prove that S is a basis. 28. Let S be the subspace of M2 (R) consisting of all 2 x 2 upper triangular matrices. Determine a basis for S, and hence, find dim[ S]. nousups 29. Let S be the subspace of M2(R) consisting of all 2 x 2 matrices with trace zero. Determine a basis for S, and hence, find dim[S]. basilammon nood and doidy 30. Let S be the subspace of R' spanned by the vectors V1 = (1, 0, 1), v2 = (0, 1, 1), v3 = (2, 0, 2). Deter- mine a basis for S, and hence, find dim[S].for Problems 19-22, determine whether the given set of vec- fors is linearly independent in P2 (R). 3 19. pi(x) = 1-x, P2(x) = 1+x. 20. PI (x) = 2+ 3x, P2(x) =4+6x. 21. PI(x) = 1 - 3x4, p2(x) = 2x + x2, p3(x) = 5