please help on these and explain iff possible each step
Let A and B be two m times n matrices and let R(A). R(B) and R(A + B) be the column spaces of A, B and A + B respectively. Prove that R(A + B) (R(A) + R(B)). Deduce that rank (A + B) le rank A + rank B. Let S', S" and U be 3 subspaces of a vector space S. Prove or disprove that S' U S" is a subspace of S. Assume that S = S' U = S" U. Prove or disprove that S' = S". Let S' = {x R3 : x3 = 0} and S" = {x R3 : x1 = 0}. Prove or disprove that S' S". Determine S' .
