I have started reading the following lecture notes on hyperplane arrangements and I am having some trouble with the essentialization of an arrangement explained at the start of the first lecture.
A set of affine hyperplanes $\mathcal{A} = \{H_1, \ldots, H_m\}$ in a vector space $V = K^n$, for some field $K$, is called essential if the dimension of the space spanned by the normals of the hyperplanes is equal to $n$, the dimension of $V$. If $K$ happens to be $\mathbb{R}$ we can let $X$ be the space spanned by the normals and consider the set of hyperplanes $\{X \cap H_1, \ldots, X \cap H_m\}$ of $X$. This is essential by definition of $X$ and the original set $\mathcal{A}$ can be recovered because $(X \cap H_i) + X^\perp = H_i$. So in this case this gives an "equivalent" set of hyperplanes which is essential, and is thus called the essentialization of $\mathcal{A}$.
This part was ok for me but the author also describes how to essentialize for a general field $K$, and that's where I cannot quite follow. The subspace $X$ is defined as before, but now we choose a complement $Y$ for it in $V$ and then take $$ W = Y^\perp = \{ v \in V : v \cdot y = 0 \; \forall y \in Y \}. $$ After that we can proceed as before with $W$ in place of $X$, that is, take the hyperplanes $\{W \cap H_1, \ldots, W \cap H_m\}$. These define an "equivalent" set of hyperplanes which is essential.
This general case is more obscure for me and I cannot quite see why (1) we obtain an essential set of hyperplanes and (2) how one can "reverse" the construction to get $\mathcal{A}$ back.
It would help me if someone could provide a clear explanation/proof for these facts, or even give me a series of simpler steps I can prove to get to those statements.
I think the key point is $codim_{W}(H\cap W)=1$. That is to say, the $rank$ of $\left\{W \cap H_{1}, \ldots, W \cap H_{m}\right\}$ is same to $rank(\mathcal{A})$ and we can take care of a lower dimension version on $W$ correspond to intersecting hyperplanes. If we want to recover the original problem, we need to add $X^{\perp}$.