This makes no sense at all. V and W are vector spaces, not linear transformations. It makes no sense to say vector spaces are "one-to-one" or "onto". Take a deep breath and think about what you are saying. It is the linear transformation L that you are to show is "one-to-one" and "onto".This would mean that V and W are one-to-one iff they are onto, and if I can use this to show that V and W are onto, then Range(L) = W.
When L is written as a matrix, "with respect to bases B for V, and C for W", you get the columns of A by applying L to each of the vectors in B in turn, writing the result as linear combination of vectors in C. That means, in particular, that if [tex]a_1, a_2, \cdot\cdot\cdot a_m[\math] is the first column, The where is the first vector in basis B and are the vectors in basis C. That is, in that sense, every column of A represents a vector in W.But how can I show that V and W are one-to-one in order to do this, and what can I do with Col(A)?
Any help would be greatly appreciated, thanks.