A principal open set in $A^n $ is an affine variety?

Question

I am trying to prove that a principal open set in $A^n$ is an affine variety in order to prove that $GL(n,K)$ is. Is there a way do this with a quite basic knowledge of algebraic geometry?

Answer 1

So a principal open subset of $\mathbb A^n$ has the form $U_f = \{ x \in \mathbb A^n \mid f(x) \neq 0 \}$.

Then define $V \subset \mathbb A^{n+1}$ by $\{ (x,t) \in \mathbb A^{n+1} \mid f(x)t-1=0 \}$. $V$ is a closed subset of affine space, so is clearly an affine variety.

Also, $U_f$ and $V$ are isomorphic via the restriction of the projection map $\pi:\mathbb A^{n+1} \to \mathbb A^n$, namely send $(x,t) \in V$ to $x \in U_f$.