Let be an F-Algebra homomorphism. I understand this to mean that and have the same algebraic structure (the meaning of which is also a little fuzzy to me…). Often we say in class “ respects the operations.”

Is it true that the pre-image of under contains (possibly multiple) pre-images of the basis vectors of ?

If so, then I have a follow up:

We often try to show that is bijective in order to establish an isomorphism. Is this necessary? Would any bijective map from the basis of to the basis of extend linearly to a bijection between the algebras? Further, would the fact that they are homomorphic (by ) and the existence of some bijection between them establish an isomorphism even though and may be unrelated?

Lastly, must any homomorphism between isomorphic objects be bijective? (Does bijective imply bijective?)

To attempt to answer your last question: two objects are isomorphic if a bijection such that , where . So this means the map $\psi$ itself must be an isomorphism (a bijective homomorphism) by definition.

Thus, this seems to suggest that . So not only must they be related, but are in fact identical.

I meant “two objects are isomorphic…”

Is this a counter-example to your reply?

Let and be , -algebras over themselves and certainly isomorphic. Define and . I think this map is bijective (though obviously not linear so not a homomorphism) and the image and pre-image are isomorphic, yet is not an isomorphism…