Hi, I am having troubles even starting this problem and any help would be appreciated.

1)The question is: If B is an nxn matrix, X = {A in M(mxn) matrix space | AB = 0} and Y = {AB | A in M(mxn) matrix space}, show that X and Y are subpaces of M(mxn) and that dim(X) + dim(Y) = mn.

What I have)From observation, I can see that X resembles a null space or kernel of a T, and Y resembles an image space or image of a T. And partially, I can see that dim(X) + dim(Y) = mn = dim(M(mxn) matrix space).

2)As a related question, can someone explain to me how toshowif a transformation is one to one and/or onto? Suppose T: V-->W: I know that one to one means that the ker(T) = {0}, does that mean when a T is one to one, there are no vectors that meet the requirement T(v) = 0? And likewise, for onto, not every vector w in W can be mapped via T(v)?

As always, thank you for any help. My main focus is the initial problem, and the explanation for one to one/onto is secondary; if there are any sources you may point me to for the explanation, please do - I just hope for a different way to approach the thinking since I am quite confused.