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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 ?

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The hint reduces you to the case $V(F_1) \cap V(x_{m+1}, \ldots x_n)$, writing $F_1 = \sum \lambda_i x_i - \lambda$, this vanishing set is just $$V(\sum_{i \leqslant m} \lambda_i x_i - \lambda) \oplus 0$$i.e. you are reduced to the base case (in the first $m$ coordinates).

To handle the base case, replace $x_1$ say with $x_1 + \frac{\lambda}{\lambda_1}$.

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