# linear algebra – Is the dimension concept always well defined for unfinished dimensional spaces?

The question is quite simple: if $$mathbb {V}$$ is a vector space and $$B$$ and $$B & # 39;$$ are the basis for $$mathbb V$$, so do $$B$$ and $$B & # 39;$$ to have the same cardinality?

I tried to answer the question as follows: suppose that $$B & # 39;$$ is taller than $$B$$. Then there is a map $$f: B & # 39; to B$$ surjective but not injective. Extend this to a linear map $$f: mathbb V to mathbb V$$ we have that $$mathbb V / ker f # cong mathrm {Im} f & # 39; = mathbb V$$with $$ker f$$ to be non-trivial. But is it really a contradiction? (Of course, it's in finite dimension, since we know that all bases have the same cardinality in this case, but that's exactly what we're trying to prove here …)