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I was considering the group of automorphisms on a vector space $\mathbb{Q}^3$ with the binary operation of addition. Is there a general description of the elements of this group? At first, it seemed to be that it would consist of bijective linear transformations, but I'm not sure how informative that is. Is there a specific category that this automorphism group would fall under?

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    @xdfm: that was very confusing. It sounded from your question as if you had already realized this; instead of "I'm not sure how informative that is" shouldn't you have said "I'm not sure how true that is"?2010-09-21

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A vector space over $\mathbb{Q}$ is the same as a torsionfree divisible abelian group. The product $p/q * v$ is the unique vector so that $q * p/q v = p v$. In particular, every group homomorphism between vector spaces over $\mathbb{Q}$ is automatically $\mathbb{Q}$-linear. Thus, for example, the group automorphisms of $\mathbb{Q}^n$ may be described by $GL_n(\mathbb{Q})$.

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The distinction between preserving addition and also preserving scalar multiplication is illusory in this case, because any vector can be divided by $n$.

The distinction resurfaces, however, for the same problem with $Q$ replaced by (almost) any other field, such as the real numbers. There preserving scalar multiplication is an additional property that does not follow from addition being conserved.

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    @yunone: multiply everything by the denominator from the start. If a = (p/n)x then nT(a)=T(na)=T(px)=pT(x).2010-09-21