# Line

Suppose $x_1, x_2$ are two points in $\mathbb{R^n}$. Then the line passing through them is defined as follows:

# Affine set

The set $A$ is called affine if for any $x_1, x_2$ from $A$ the line passing through them also lies in $A$, i.e.

## Examples:

• $\mathbb{R}^n$
• The set of solutions $\left\{ x \mid \mathbf{A}x = \mathbf{b}\right\}$

# Related definitions

## Affine combination

Let we have $x_1, x_2, \ldots, x_k \in S$, then the point $\theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_k x_k$ is called affine combination of $x_1, x_2, \ldots, x_k$ if $\sum\limits_{i=1}^k\theta_i = 1$

## Affine hull

The set of all affine combinations of points in set $S$ is called the affine hull of $S$:

• The set $\mathbf{aff}(S)$ is the smallest affine set containing $S$.