A linear equation solver using gaussian elimination. Implemented for fun and learning/teaching.

The solver will solve equations of the type:

A can be rectangular and/or singular. The solver will return a particular solution as well as a list of vectors spanning the null space of A required for the general solution.

The solution is acquired following these steps:

1. The solver creates the augmented matrix [A | b].
2. The augmetned matrix is reduced to row-echelon form (REF) using Gaussian elimination.
3. If the number of pivots are fewer than the number of nonzero entries in REF b, the solver exits.
4. The augmented matrix is then further reduced into a reduced row-echelon form (RREF).
5. The solver reads off the particular from the RREF.
6. Finally it finds the null space using back-substitution on the RREF.

A = np.array([
    [-2, 4, -2, 1, 4],
    [4, -8, 3, -3, 1],
    [1, -2, 1, -1, 1],
    [1, -2, 0, -3, 4]
])

b = np.array([-3, 2, 0, 1])

solver = Solver(A, b)
xp, null_space = solver.solve()