I'm trying to prove that if $V$ is a non-empty linear subvariety then there is an affine change of coordinates $T$ of $ \Bbb A^n $ such that $V^T = V(X_{m+1}, \ldots, X_n) $. A set V in $ \Bbb A^n(k) $ is called a linear subvariety of $ \Bbb A^n(k) $ if $V = V (F_1, \ldots ,F_r )$ for some polynomials $F_i$ of degree $1$.
I found this on the book Algebraic Curves by William Fulton. Note that in the book there is a hint : (use induction on $r$).
Do you have some other suggestions ?