main.py

import numpy as np

def gaussian_elimination(A, b):
    """
    Gaussian elimination is a method for solving systems of linear equations. It involves transforming a matrix into row echelon form through a series of elementary row operations. Key points:

1. Used to solve linear systems: $Ax = b$
2. Steps:
   - Convert to augmented matrix [A|b]
   - Transform to row echelon form
   - Back-substitute to find solutions

3. Complexity: $O(n^3)$ for an $n \times n$ matrix
"""
    n = len(b)
    for i in range(n):
        A[i] = np.append(A[i], b[i])
    for i in range(n):
        A[i] = A[i] / A[i][i]
        for j in range(i + 1, n):
            A[j] = A[j] - A[i] * A[j][i]
    x = np.zeros(n)
    for i in range(n - 1, -1, -1):
        x[i] = A[i][-1] - np.dot(A[i][i:-1], x[i:])
    return x