Transformation T is... "onto"? I thought you have to say a mapping is onto something... like, you don't say, "the book is on the top of"...
Our book starts out by saying "a mapping is said to be onto R^m", but thereafter, it just says "the mapping is onto", without saying onto what. Is that simply the author's version of being too lazy to write the codomain (sorry for saying something negative, but that's what it looks like to me at the moment), or does it have a different meaning?

You do indeed hear these terms in relation to functions.
One-to-one means the same as injective.
Onto means the same as surjective.
One-to-one and onto means bijective.
A function can be just one of them or all three of them.
To answer your specific question, onto means each value of the codomain is mapped to by a member of the domain.
