injective vs. surjective

Last semester, I spent a lot of time beating my head against the wall of isomorphisms and how to prove that something was injective or surjective, which was addressed in class but assumed in the text.  After twenty minutes with Shedon Axler's Linear Algebra done right, I actually understood (at least momentarily) how to write a proof of injectivity versus surjectivity**.  I must confess after a year of acquaintance with these terms I still have to keep looking them up to get them straight.  I end up having to use a silly mnemonic to remember: the first letter of injective, an i, looks like the numeral 1, so it means 1-to-1.

injective (1-to-1) but not surjective (does not reach whole codomain Y) - image from wikipedia

surjective but not injective - image from wikipedia

Anyway, for whatever it's worth - mostly my own reference - here are few notes on proving something is injective or surjective:

To prove a transformation T: X->Y is injective, you show that for v, u in X, if Tv=Tu, then v=u.   It is equivalent to show that the only element of the nullspace is {0}.  To prove something is surjective, show that the range of a transformation is the entire codomain Y.

 I'm embarrassed that I have to keep looking all this stuff up, as it is a very basic building block for proofs, but maybe with enough repetition it will eventually stick in my brain.