Section 1 Definitions

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

(1)
\begin{equation} a_1 x_1 + a_2 x_2 + ... + a_n x_n = b \end{equation}

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

(2)
\begin{align} y=c_1\bar{v}_1 + c_2\bar{v}_2 + ... + c_p \bar{v}_p \end{align}

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

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