 Scalar kernel of the projective linear action

A linear transformation fixing every one-dimensional subspace is scalar. Fixing the coordinate lines makes it diagonal, and fixing the lines spanned by $e_i+e_j$ makes all diagonal entries equal. Within determinant-one matrices the scalar satisfies ${\lambda }^n=1$. Conversely scalars fix all projective points. Commutation with all elementary transvections shows these scalars are exactly the center of the special linear group, so its projective quotient acts faithfully.