Construction of the Associated projective hyperplane. Let $V$ be $K-$vector space of dimension $n$. Let $H$ be a hyperplane given by the kernel of the linear map
$l_H:V\longrightarrow K$ defined by
$$l_H(x)=a_1x_1+\cdots+a_nx_n$$
where $x_1,\cdots,x_n$ are the coordinates of $x$ in a chosen basis of $V$.
Let $P(V)$ be the projective space associated to $V$. Consider the map: $l_{H'}:P(V)\longrightarrow K$
defined by
$$l_{H'}([x_1:\cdots:x_n])=a_1x_1+\cdots+a_nx_n$$
Does the kernel of $l_{H'}$ defines a projective hyperplane in $P(V)$?
The answer is yes. A point $M \in P(V)$ belongs to the projective hyperplane $P(H)$ if and only if one of its representative, $\vec{M} \in H \iff l_H(\vec{M}) = 0$ (which is an equation of $P(H)$.
The answer is yes. A point $M \in P(V)$ belongs to the projective hyperplane $P(H)$ if and only if one of its representative, $\vec{M} \in H \iff l_H(\vec{M}) = 0$ (which is an equation of $P(H)$.