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 system of linear equations has 1) no solution, or 2) exactly one solutions, or 3) infinitely many solutions.

A **solution** of the system is a list of numbers that makes each equation true. If the matrix has a solution, it is considered **consistent**, if not, it is inconsistent.

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

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

In a matrix, a vertical dashed line indicates the other side of an equation. This is a way of denoting an augmented matrix versus a variable coefficient matrix. While this is not a normal idea in math, just Dr. Villalpando's idea, it should be the normal.

#### 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 $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$