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

\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 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.

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


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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