A set $V\subset \mathbb{P}^n$ is called a linear subvariety of $\mathbb{P}^n$ if it's the zero locus of $r$ homogeneous and linear, i.e $V=Z(H_1,...,H_r )$ where each $H_i$ is a form of degree 1.

I want to know if there is a simple way to prove the following:

Prove that there exist a projective linear change of coordinates $T:\mathbb{P}^n \to \mathbb{P}^n$ such that $T^{-1}V=Z(x_{m+2},...,x_{n+1})$. And $m$ is the dimension of $V$.

This is what I did:

Let's assume that the $H_i$ are LI over $k$. Observe that $T^{-1}V$ is again a linear subvariety and is given by $T^{-1}V=Z(H_1\circ T,...,H_r\circ T)$. We look for such $T$. The transformation is given by $T=(T_1,...,T_{n+1})$ and each $T_k(x_1,...,x_{n+1})=\sum_{j=1}^{n+1}{a_{kj}x_j}$. We can solve a giant system an prove that there exist solution for such $a_{kj}$ under some conditions and amounts of equations.....

I think that this is not the correct way

Maybe interpreting the problem on the affine space $\mathbb{A}^{n+1}$ would help but I don't know how.

Please help me, I don't want to find explicitly such transformation, just prove existence. Thanks!


You're just supposed to observe that there's a linear transformation taking any subspace to any other of the same dimension. One way to see this is to note that the matrix

$$A = \left(\begin{array}{c|c|c|c} & & & \\ v_1 & v_2 & \cdots & v_n \\ & & & \end{array}\right)$$

gives $A e_i = v_i$, so we also have $A^{-1} v_i = e_i$. Using this idea it's easy to see that you can make a matrix moving any given subspace to any other of the same dimension.


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