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?

As I mentioned in my comment, the word "onto" is often used as a synonym for the word "surjective". In the same spirit, you can use "one-to-one" instead of "injective". See for example the corresponding Wikipedia article.

**Edit** : I agree with the comments by Qiaochu and Jonas that "one-to-one" is a little ambiguous and could refer to a bijection. So it is probably best to stick to the unambiguous terms "injective" and "surjective".