(For completeness, I assume we are dealing with vector spaces of finite dimension.)

If T is a bijective linear transformation, then it must necessarily be so that dim(V)=dim(W). The converse implication is not generally true, that is, even though we know that dim(V)=dim(W), the linear transformation may not be bijective.

To see why this is so, you can take V=W=R (where R is the real line), so V and W are both vector spaces of dimension 1. Then the map T : V -> W defined by T(v) = 0 for all v in V is a linear transformation, and it is certainly not a bijection.

If you have a more specific context, maybe we can help further.