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Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:

Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!

I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help.

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    The image of this function is not in $F^n$2012-11-24
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    Are you aware that any vector space of dimension $n$ is isomorphic to $F^n$? "As abelian groups" means that you ignore scalar multiplication and just look at addition.2012-11-24
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    I think that you meant to ask about finite dimensional (as opposed to finite) vector spaces, so I edited the question accordingly. Do roll back, if I misunderstood. The only finite vector space over $\mathbb{R}$ is the 0-dimensional space, and I doubt the question wanted you to concentrate on that. Over finite fields there would be other finite vector spaces, but it sounds like you are not restriced to a finite $F$ (and the result holds for all fields anyway).2012-11-24
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    Your title doesn't match your question. You should change whichever is wrong.2012-11-24
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    @JyrkiLahtonen Sorry, I'm translating the question to English and I might have done so incorrectly. It does make more sense for $V$ to be finite dimensional.2012-11-24
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    @ChrisEagle Sorry, I don't see where is the mistake, could you please be more specific?2012-11-24
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    @Amr I was thinking of $F^n$ as the direct sum of $F$ rather than direct product in my first attempt, then my function is well-defined, no? It's of no use though.2012-11-24
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    @Zero: In your title you speak of a vector space being isomorphic to a field, in your question you speak of it being isomorphic to $F^n$, where $F$ is a field.2012-11-24
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    @ChrisEagle would you be able to take a look at this? https://math.stackexchange.com/questions/2374069/generalization-of-n-th-dimensional-vector-space-isomorphic-to-n-th-product-o I'm trying to generalize what Zero is asking about here.2017-07-27

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