How to go from $GL(V)$ to $GL_n(F)$?
During my exams, I was asked what is the difference between $GL(V)$, $GL_n(F)$, $GL_n(R)$, $GL_n(\mathbb{R})$ etc. and how could one go for example from $GL(V)$ to $GL_n(F)$ and opposite. (It was probably meant as how does one obtain one type of general linear group from the other.)
I think the groups are defined like this:
- $GL(V)= \{ f: V \rightarrow V | \text{f is bijective and linear} \}$ = abstract automorphism group, not necessarilly with elements = matrices.
- $GL_n(F)$ = $n \times n$ invertible matrices with elements in a field $F$ (or a ring $R$ or real numbers in the other cases).
However, I am not sure for example how does one transform $GL(V)$ so it becomes $GL_n(F)$. Are there any extra conditions to put on $GL(V)$, e.g. somehow turning the $f$ elements from $GL(V)$into matrices? Or am I missing the point completely?
Thank you for your advice.