Linear Algebra

import numpy as np
A = np.array([[1,2],[3,4]])
B = np.array([[5,6],[7,8]])

print(A @ B)          # matrix multiply (recommended)
print(np.dot(A, B))   # same as @
print(A * B)          # element-wise (NOT matrix mul!)

v = np.array([1, 2])
print(A @ v)          # [5, 11] matrix-vector

A = np.array([[1,2],[3,4]])
print(np.linalg.det(A))       # -2.0
print(np.linalg.inv(A))       # [[-2, 1],[1.5,-0.5]]
print(A @ np.linalg.inv(A))   # identity (verify)
print(np.linalg.matrix_rank(A))  # 2
print(np.trace(A))            # 5 (sum of diagonal)

# Eigendecomposition
vals, vecs = np.linalg.eig(A)
print(f"Eigenvalues: {vals}")   # [-0.372, 5.372]
print(f"Eigenvectors:\n{vecs}")

# SVD (used in PCA, recommendation systems)
M = np.array([[1,2],[3,4],[5,6]])
U, S, Vt = np.linalg.svd(M, full_matrices=False)
print(f"U shape: {U.shape}")    # (3,2)
print(f"S: {S}")                # singular values
print(f"Vt shape: {Vt.shape}")  # (2,2)

# Solve Ax = b
A = np.array([[2,1],[5,7]])
b = np.array([11, 13])

x = np.linalg.solve(A, b)
print(f"Solution: {x}")  # [7.111, -3.222]
print(np.allclose(A @ x, b))  # True (verify)