Let \(X, Y\subset\mathbb A^ n\) be non-empty closed subvarieties of affine space \(\mathbb A^ n\) over an algebraically closed field of characteristic zero. Let \(D_ X\), \(D_ Y\) be the Lie algebra of global vector fields on \(X\), respectively on \(Y\). The authors prove that each isomorphism between \(D_ X\) and \(D_ Y\) is induced by an algebraic automorphism of \(\mathbb A^ n\) mapping \(X\) onto \(Y\).