Chapter 160 — Systems of Linear Equations

# --- Systems of Linear Equations: Three cases ---
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn-v0_8-whitegrid')

def classify_system(A, b):
    """
    Classify Ax=b: unique, no solution, or infinite solutions.

    Args:
        A: 2D array (m, n)
        b: 1D array (m,)

    Returns:
        string classification
    """
    m, n = A.shape
    rank_A = np.linalg.matrix_rank(A)
    # Augmented matrix [A | b]
    Ab = np.column_stack([A, b])
    rank_Ab = np.linalg.matrix_rank(Ab)
    if rank_A < rank_Ab:
        return f"No solution (inconsistent): rank(A)={rank_A} < rank([A|b])={rank_Ab}"
    elif rank_A == n:
        return f"Unique solution: rank(A)={rank_A} = n={n}"
    else:
        return f"Infinite solutions: rank(A)={rank_A} < n={n}, {n-rank_A} free variables"

# Test cases
cases = [
    (np.array([[2.,1.],[1.,3.]]), np.array([5., 7.])),           # unique
    (np.array([[1.,2.],[2.,4.]]), np.array([3., 7.])),           # no solution
    (np.array([[1.,2.],[2.,4.]]), np.array([3., 6.])),           # infinite
    (np.array([[1.,0.,1.],[0.,1.,1.],[1.,1.,2.]]), np.array([1.,2.,3.])),  # infinite 3D
]
for A, b in cases:
    print(classify_system(A, b))

# 2D visualization: lines as rows of Ax = b
fig, axes = plt.subplots(1, 3, figsize=(14, 4))
x = np.linspace(-2, 6, 300)

# Unique: two lines intersecting
A1 = np.array([[2.,1.],[1.,3.]]); b1 = np.array([5.,7.])
sol = np.linalg.solve(A1, b1)
axes[0].plot(x, (b1[0]-A1[0,0]*x)/A1[0,1], 'steelblue', lw=2, label='Eq 1')
axes[0].plot(x, (b1[1]-A1[1,0]*x)/A1[1,1], 'crimson', lw=2, label='Eq 2')
axes[0].scatter(*sol, color='black', zorder=5, s=80, label=f'Solution {sol.round(2)}')
axes[0].set_xlim(-1,5); axes[0].set_ylim(-1,5); axes[0].legend(); axes[0].set_title('Unique solution')

# No solution: parallel lines
A2 = np.array([[1.,2.],[2.,4.]]); b2 = np.array([3.,7.])
axes[1].plot(x, (b2[0]-A2[0,0]*x)/A2[0,1], 'steelblue', lw=2, label='Eq 1')
axes[1].plot(x, (b2[1]-A2[1,0]*x)/A2[1,1], 'crimson', lw=2, linestyle='--', label='Eq 2')
axes[1].set_xlim(-1,5); axes[1].set_ylim(-2,4); axes[1].legend(); axes[1].set_title('No solution (parallel)')

# Infinite: same line
A3 = np.array([[1.,2.],[2.,4.]]); b3 = np.array([3.,6.])
axes[2].plot(x, (b3[0]-A3[0,0]*x)/A3[0,1], 'steelblue', lw=4, alpha=0.4, label='Eq 1 (same line)')
axes[2].plot(x, (b3[1]-A3[1,0]*x)/A3[1,1], 'crimson', lw=2, linestyle='--', label='Eq 2')
axes[2].set_xlim(-1,5); axes[2].set_ylim(-1,3); axes[2].legend(); axes[2].set_title('Infinite solutions (same line)')

for ax in axes: ax.axhline(0,color='gray',lw=0.5); ax.axvline(0,color='gray',lw=0.5)
plt.suptitle('Three Cases for Ax = b', fontsize=13); plt.tight_layout(); plt.show()