# How does the following definition of isomorphism between vector space imply “structure preserving bijection”?

I am really confused by this concept of isomorphism, it seems to be a new name for something that is already well understood. Every time I look up the definition for isomorphism, the definition changes. In other fields, they are called homomorphism, which confuses with homeomorphism (let's not get into that). They are of course equivalent, but the argument is quite subtle.

Acccording the following reliable source (Naylor and Sell):