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.
The projective linear group is indeed an affine variety and all affine varieties are closed subsets of some affine space (defined by polynomials). Any smooth affine variety $X$ can be embedded in an affine space of dimension $2\dim X+1$, though for $PGL_n$, the natural embedding is in something much larger. Let $V$ be the vector space of all degree $n$ monomials in $n^2$ variables. Then there is a natural embedding of $PGL_n$ in $V$, given by a point in $PGL_n$ represented by $n^2$ coordinates, to all monomials of degree $n$ in the coordinates divided by the determinant of the matrix. It lies inside the hyperplane given by $\det =1$. The equations can be theoretically written down and usually known as the Veronese equations, since this is like the Veronese embedding of projective space using degree $n$ monomials.