That is, if u and v are in X you must prove that u+ v is in X. If u is in X then f(u) is in Y. If v is in X then f(v) is in Y. What can you say about f(u+ v)?
Similarly, if u is in X then f(u) is in Y. What can you say about f(au) for a any scalar?
b) Prove that if f is surjective and V is finite dimensional, then dim V - dim X = dim W - dim Y.
Here I am lost. I understand (or I think I do) surjectivity, and I think that the restriction of f to X gives a linear transformation g: X -> W with g(x) = f(x) for all x (an element of) X, which I'm fairly sure is relevant to the solution, but again, I have no idea with how to go about setting out a formal proof.
I apologise for the "(an element of)", I'm still learning the formatting. Thank you for your time.[/QUOTE]