A **linear equation** in the variables $x_1, x_2, ... , x_n$ is an equation that can be written in the form

A **system of linear equations** is a collection of linear equations involving the same variables.

A **solution** of the system is a list of numbers that makes each equation true.

The set of all solutions is the **solution set**.

Two systems are **equivalent** if they have the same solution set.

#### Linear Combination

Let $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$ be vectors in $\mathbb{R}^n$.

Let $c_1, c_2, ..., c_p$ be scalars.

Then

is the linear combination of $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$

#### Span

Let $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p \in \mathbb{R}^n$. The **span** of $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$ is the set of all linear combinations of$\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$