$\newcommand{\C}{\mathbb{C}}$ Let the underlying field be $\C$.
As noted in my previous answer there is always an embedding of $PGL(k+1)$ into $GL(n)$ where $n$ depends on $k$: We have the adjoint representation $PGL(k+1) \to GL(Lie(PGL(k+1)))$ sending $A \in PGL(k+1)$ to $ad_{\tilde A}$ where $\tilde A$ is any lift to $GL(k+1)$.
I am trying to find the minimal $n$ that will work for each $k$. As a start, why aren't there any nontrivial homomorphisms of $PGL(k+1)$ into $GL(k+1)$? I am trying to find an answer using schur multipliers but I can't seem to get my hands on any reference about this.
e.g. I hope to use the fact that $H_2(BPGL(n))=H_2(Gr_n^\infty) \neq 0$.