Linear solver
A is a square matrix n-by-n and `\vecu` a vector of dimension n.
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]] = [[5],[0],[-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) = [[5],[0],[-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.