I have read this question. Also, there are theorems telling me that $PGL_n$, as the quotient of $GL_n$ by its center, is with no doubt an affine variety (affine algebraic group).
But, is it true that every finite dimensional affine variety can be viewed as a closed subset of an affine $m$-space $\mathbb A^m$? Can $PGL_n$ be viewed as a closed subset of $\mathbb A^m$ for $m \geq \dim PGL_n$? If this is true, then what are the defining polynomials of $PGL_n$ in $k[x_1,x_2,\cdots,x_m]$?
Thanks to everyone.