= Scalar kernel of the projective linear action
{title2=$\ker[SL_n(F)\curvearrowright\mathbb P^{n-1}(F)]=\{\lambda I:\lambda^n=1\}$}
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.
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