[This was intended to be comment to Ben's reply but I exceeded the allowable limit for comments.]
Actually it doesn't work over any ring. Just take any ring $R$ for which 
$GL(V)(R)  \to PGL(V)(R)$ is not surjective. This exists if $R$ has a non-trivial rank $1$ projective $L$ such that $L^n$ is isomorphic to $R^n$. Then $V\bigotimes L$ is an $End(V)\bigotimes R$ module which is not isomorphic to $V$ though they are both indecomposable projective $End(V)\bigotimes R$-modules (rarely irreducible though). The factor that the sum of $n$ copies of them are isomorphic gives an automorphism 
of $End(V)\bigotimes R$ (which can be taken as the definition of the $R$-points of $PGL(V)$) that does not lift to an element of $GL(V\bigotimes R)$. As projective modules over a local ring are free the $GL(V)(R)  \to PGL(V)(R)$ is surjective when $R$ is local which is enough to show that $GL(V) \to PGL(V)$ is surjective as a map of algebraic groups.