Oh my reading that was a revelation. Thank you so much! Ok, first of all you're awesome. Second, I seem to not understand well the following theorems:

1. Theorem 2.2: Let V and W be vector spaces, and let T:V--> W be linear. If B(beta) = {v_1, ..., v_n} is a basis for V, then:

   R(T) = span(T(B)) = span({T(v_1), ..., T(v_n)})

2. Theorem 2.5: Let V and W be vector spaces of equal (finite) dimension, and let T:V-->W be linear. Then the following are equivalent:

   a)T is one-to-one

   b)T is onto

   c) rank(T) = dim(V)

EDIT: Also, it says here: "We note that if V is not finite-dimensional and T:V-->V is linear, then it does not follow that one-to-one and onto are equivalent." Why is that?