Linear System

A linear system is a finite collection of Linear Equations. Quite simply, a linear system is just a set of linear equations that involve the same variables. Augmented Matrices are a useful way to represent linear systems.

Homogeneity §

A linear system is considered homogenous if all the linear equations in the system have no constants , i.e. all the variables can be put on one side of the equation, equalling 0.

$$\{\begin{array}{l}x+y-z=0\\ 3x=4z\end{array}$$

The system above is homogenous, as it has no constant terms, only variables and coefficients.

Solutions §

The solution of a linear system of the variables $x_1,x_2,x_3\dots x_n$ is a set of values that satisfy every equation in the system:

$$\{\begin{array}{l}2x=8\\ x+y=8\end{array}$$

The following linear system of $x,y$ has a solution: $x=4$ and $y=4$. However, not every linear system has solutions.

Whether a system has a solution (or multiple solutions) or not is determined by it’s Consistency:

Applications §

Polynomial Interpolation §

Given enough points, a polynomial function $y=a{x}^n+b{x}^{n-1}+\dots c$ can be constructed by determining the values of $a,b,c\dots$ using the points.