1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If A is m x n, then A and (A^T)(A) have the same rank. 1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T). [I believe it's true that AX is in null(A^T), but I am totally unsure whether AX is in col(A) or not!?] 1c) If U and W are subspaces of a vector space V, then the set of vectors that belong to either U or W is a subspace of V. 2) Prove that if A is an m x n matrix, then null(A)=[col(A^T)]^| [Using dimension theorem, I proved that their dimensions are equal...but I have no idea how to prove that they ARE equal...] These are also the past exams questions that I am having terrible trouble with. Can someone give me some advice/hints? For 1b) and 2), I am partially done, but how 1a)c) I have no clue... Any help/hints is greatly appreciated!