Line

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$ and $x_{1}, x_{2} \in V$. The Line $L \subset V$ containing $x_{1}$ and $x_{2}$ is the Set $$L = {\lambda x_{1} + (1 - \lambda) x_{2} : \lambda \in \mathbb{R}}$$

1. $L$ can also be expressed as $L = {x_{2} + \lambda (x_{1} - x_{2}) : \lambda \in \mathbb{R} }$. This follows from the observation that $$\lambda x_{1} + (1 - \lambda) x_{2} = x_{2} + \lambda x_{1} - \lambda x_{2} = x_{2} + \lambda (x_{1} - x_{2})$$

• Real Numbers