Let V and W be linear spaces onto field F.

T:V W is called Isomorphism from V over W iff it has these three properties:

1) T is a linear transformation

2) T is one-to-one

3) T is onto W

So by that definition of Isomorphism can I say that if KerT has more than one vector that transforms to then those subspaces are not isomorphic.

Basically, I'm concluding that if two spaces are isomorphic to each other, only the zero vector of V transforms to the zero vector of W