# Fulton, algebraic curves exercise 4.11

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem:

A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if $V=V(H_1,\dots,H_r)$, where each $H_i$ is a form of degree $1$.

a) Show that if $T$ is a projective change of coordinates, then $V^T=T^{-1}(V)$ is also a linear subvariety.

b) Show that exists a projective change of coordinates $T$ of $\mathbb{P}^n$ such that $V^T=V(x_{m+2},\dots,x_{n+1})$, then $V$ is a variety.

c) Show that the $m$ that appears in part (b) is independent of the choice of $T$.

Can anyone give me a Hint to do the part (b) and (c)?

Any help will be appreciated!

Hint: Linear Algebra. Namely, hyperplanes are the zero locus of linear forms that can be regarded as elements of the dual vector space $(k^{n+1})^*$. Keep in mind the theorem in linear algebra that claims that given a set $\lambda_1, \cdots, \lambda_r$ of linearly independent $1$-forms then there exists another $n+1-r$ $1$-forms $\lambda_{r+1},\cdots,\lambda_{n+1}$ such that all together $\lambda_1,\cdots,\lambda_{n+1}$ are a basis of $(k^{n+1})^*$. If the hint is not sufficient for you I can add more details.