If $F$ is a field with $q$ elements, $V$ a $F$-vector space of dimension $n$, then how many ordered basis does $V$ have ?
First, $V$ has $q^n$ elements, am I correct ? Let $\{v_1,\dots,v_n\}$ be a basis then for example $\{v_1+v_2,v_2,\dots,v_n\}$ is also a basis, So can I say $n\times n$ identity matrix represents the first basis, then for the second basis we have also the identity matrix but with the first row replaced by $(1,1,0,\dots,0)$ (instead of $(1,0,0,\dots,0)$), hence any invertible matrix would give another basis, but it should be ordered, does it mean that the matrix should not be symmettric (axis ?) and don't we need some information about $q$, for example $q=p^n$ then number of elements depend on $p$ ?
