[Math] Automorphism group of vector space

group-theoryrepresentation-theoryvector-spaces

I was trying to understand definition of representation and trivial representation thus came across the case where $ V= K $ here $V$ is a vector space over a field $K$
and thus $Aut_K (V) \cong K^{*}$ but I am not quite clear about this automorphism groups.
I understand that we need invertible etc. but can someone explain this group ?
the elements of this group.

EDIT ( another Q) :

and also if we have dim$(V)$ = $n$ then we can say $Aut_K (V) \cong GL_{n} (K) $ is it just because every map $f \in Aut_K (V) $ can be expressed as $ [f]_{\beta}$ for some $\beta \in K$ to give $[f]_{\beta} \in GL_{n} (K) $ is this true ?

Best Answer

$K^*$ is the group of all non zero elements of the field $K$ under multiplication. Basically, all this is saying is that all automorphisms from a field to itself is given by multiplication by a non zero element of the field. To see why this is true, show that any such automorphism is determined by the image of a certain special element.

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