I'm taking a linear algebra course and I haven't needed to do proofs before this class (I'm an EE major and we take "special" math for Calc and Diff Eq). I could really use some help on the proofs for vector spaces. I have a quiz on Tuesday and one of these proofs will be on there. I know these would be consider simple but they are giving me a lot of trouble.
1. Let A and B be row equivalent matrices.
(a) Show that the dimension of the column space of A equals the dimension of the column space of B.
(b) Are the column spaces of the two matrices necessarily the same? Justify your answer.
2. Let A be an m x n matrix with rank equal to n. Show that if x ≠ 0, and y=Ax, then y ≠ 0.
3. Let A be an m x n matrix.
(a) If B is a nonsingular m x n matrix, show that BA and A have the same nullspace and hence the same rank.
(b) If C is a nonsingular n x n matrix, show that AC and A have the same rank.
4. Let A ∈ R^(m x n), B ∈ R^(n x r), and C = AB. Show that
(a) The column space of C is a subspace of the column space of A.
(b) The row space of C is a subspace of the row space of B.
(c) rank(C) ≤ min(rank(A), rank(B)).