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# 01 System of linear equations.
A linear equation is anything of the form $$
c_{1} x_{1} + c_{2} x_{2} + \cdots +c_{n} x_{n} = d
$$where each of the $c_{i}$ and $d$ are **scalars** (**numbers**), and each of the $x_{i}$ is an unknown **variable** (or **indeterminant**).
Notice now each variable $x_{i}$ is by itself, a monomial of degree 1. It is not being multiplied to any other variables, or in any other functions.
So for example $$
3x+2y-z=2
$$is an equation that is linear in $x,y,$ and $z$.
The equation $$
3x^{2}+2yz-\sin(z)=2
$$is not linear in $x,y,z$. It is however linear in $x^{2},yz,$ and $\sin(z)$.
To add a bit of terminology, expressions of the form $$
c_{1} x_{1} + c_{2} x_{2} + \cdots + c_{n}x_{n}
$$is a **linear combination** in the objects $x_{1},x_{2},\ldots,x_{n}$, with coefficients $c_{1},c_{2},\ldots,c_{n}$ (the scalars).
A **system of linear equations** is just a collection of linear equations:$$
\left\{
\begin{array}{}
c_{11}x_{1} + c_{12} x_{2} + \cdots + c_{1n}x_{n} = d_{1} \\
c_{21}x_{1} + c_{22} x_{2} + \cdots + c_{2n}x_{n} = d_{2} \\
\vdots \\
c_{k1} x_{1}+c_{k2}x_{2} +\cdots+c_{kn}x_{n}=d_{k}
\end{array}
\right.
$$Above shows a system of linear equations, consisting of $k$ equations and $n$ variables, each equation is linear over these $n$ variables. Sometimes we will refer to it as a **linear system** for short.
Note it is often desired to organize and align the same variables in the same column.
Now the immediate question is, how do we solve for the unknown variables in such a linear system?