The above proposition shows that a straight line is a solution set of a linear equation with two unknowns. In other words, given a point $A=(\alpha_1,\alpha_2)$ in a number plane $\mathbb R^2$ which is not the origin and a number $b\in\mathbb R$, a straight line $L$ is a subset of all points of the plane $\mathbb R^2$ defined as

$$L:=\{P\in\mathbb R^2:\; P=(x_1,x_2): \alpha_1x_1+\alpha_2x_2=\beta\}.$$

This concept can be generalized.

Definition: Hyperplane of a Number Space

For $n\ge 1$, given a point $A\in\mathbb R^n$ in the number space $\mathbb R^n$ which is not the origin, as well as a number $b\in\mathbb R$, a hyperplane $H$ is the set of solutions of the linear equation with $n$ unknowns:

$$H:=\{P\in\mathbb R^n:\; P=(x_1,\dots,x_n): \alpha_1x_1+\ldots+\alpha_nx_n=\beta\}.$$

Explanations: 1

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  1. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013