# Hyperplane

A hyperplane in an $n$-dimensional vector space $V$ over a field $\mathbb{F}$ is an $n-1$ dimensional subplane. This is given by $n-1$ linearly independent vectors $v_1, v_2, \ldots, v_{n-1} \in V$ and a point $x \in V$. Then the space is made of

$$ P = \{x + \sum_{i=1}^{n-1} \alpha_i v_i \mid \alpha_i \in \mathbb{F}\} $$

Note: different vectors $v_1, \ldots, v_{n-1}, x \in V$ could define the same hyperplane.