# Linear solver

Resolves for vector \vecx equation A.\vecx = \vecu
A is a square matrix n-by-n and \vecu a vector of dimension n.

This tool is a linear solver which solves equations of type A.\vecx = \vecu with A, a square matrix, \vecu, a given vector and \vecx, the unknowns vector.
Any linear system with n linear equations and n unknowns can be transformed into a linear solver that can be solved with this tool.

## What is the linear solver ?

This solver is used to solve a system of n linear equations and n unknowns which can be represented by in a matrix form.

Example of a system of 3 equations and 3 unknowns :

x_1 + 2x_2 + 6x_3 = 5
-x_1 + 5x_2 + x_3 = 0
6x_1 -9x_2 + 4x_3 = -2

This system can be rewritten in the following matrix form:

[[1,2,6],[-1,5,1],[6,-9,4]] . [[x_1],[x_2],[x_3]] = [,,[-2]]

We can write the correspondant linear solver as follows,

A.\vec(x) = \vec(u)

where,

A= [[1,2,6],[-1,5,1],[6,-9,4]]

\vec(x)= [[x_1],[x_2],[x_3]]

\vec(u) = [,,[-2]]

How to solve a system of n equations and n unknowns

One method is to write the system in a matrix form as described above i.e., A.\vec(x) = \vec(u)

A is a square matrix of order n.

\vec(u) is a vector of dimension n.

\vec(x) is a vector of dimension n considered of the unknown n of the system.

A single solution exists if and only if the A matrix is invertible and the solution can be written,

\vec(x) = A^(-1) . \vec(u)

With this method, solving a system of n equations and n unknowns implies to inverse a square matrix of size n.