  We introduce and study the notion of affine varieties associated to ordered
bases and establish Galois connection between the power set of $A^n_K$ and the
power set of $K[x_1, . . ., x_n]$, and then induce a Galois correspondence. We
generalize the idea by defining affine varieties associated to linear
operators. We produce Hilbert's Nullstellensatz version for such varieties and
show that there is a 1-1 correspondence between this kind of varieties in
$A^n_K$ and the "usual" affine varieties in $A^n_K$. We prove that the \usual"
affine varieties forms a skeleton for the category of all affine varieties
associated to linear operators, and hence they are equivalent categories.
