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Proposition: Presentation of a Straight Line in a Plane as a Linear Equation

Let $L$ be a straight line in the number plane $\mathbb R^2.$ Then

• $L$ goes through the origin $(0,0)\in L$ if and only if $L$ is the set of solutions of the homogenous linear equation $\alpha_1x_1+\alpha_2x_2=0$ for some real-valued coefficients $\alpha_1,\alpha_2\in\mathbb R$ which do not both equal zero.
• $L$ does not go through the origin $(0,0)\not\in L$ if and only if $L$ is the set of solutions of the inhomogenous linear equation $\alpha_1x_1+\alpha_2x_2=\beta$ for some real-valued coefficients $\alpha_1,\alpha_2\in\mathbb R$ which do not both equal zero and some real number $\beta\in\mathbb R$, $\beta\neq 0.$

Table of Contents

Proofs: 1

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Definitions: 1
Explanations: 2

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References

Bibliography

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