Hyperplane

Let $a_1$, $a_2$, ..., ${a_n}$ be Scalars not all equal to 0. Then the Set $S$ consisting of all Vectors

$${\bf X} = \left[{\matrix{x_1\cr x_2\cr \vdots\cr x_n\cr}}\right]$$

in $\Bbb{R}^n$ such that

$$a_1x_1+a_2x_2+\ldots + a_nx_n = 0$$

is a Subspace of $\Bbb{R}^n$ called a hyperplane. More generally, a hyperplane is any co-dimension 1 vector Subspace of a Vector Space. Equivalently, a hyperplane $V$ in a Vector Space $W$ is any Subspace such that $W/V$ is 1-dimensional. Equivalently, a hyperplane is the Kernel of any Nonzero linear Map from the Vector Space to the underlying Field.