Systems of Linear Equations

We seek values $x_1,x_2,\dots ,x_n$ that satisfy multiple linear equations. Linear equations have no powers (exponents) on $x$, no products on $x$.

We need the same number of equations and unknowns ($n\times n$ matrix for $[A]$).

General form:

$$\begin{array}{r}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\\ \dots \\ a_{n1}{x}_1+a_{n2}{x}_2+\cdots +a_{nn}{x}_n=b_n\end{array}$$

where

$$\begin{array}{rl}{a}_i,b_i& =\text{constants}\\ x_i& =\text{unknowns}\\ n& =\text{\# of equations, \# of unknowns}\end{array}$$

In matrix form:

$$[A{]}_{n\times n}\{X{\}}_{n{\times }_1}=\{B{\}}_{n{\times }_1}$$

where $\{\}$ is used to denote a column vector.