الجبر الخطي لتعلم الآلة

import numpy as np

# ── Vectors and dot products ──────────────────────────────────────────────────
a = np.array([3., 4.])
b = np.array([1., 0.])

print(f"a·b = {np.dot(a, b):.2f}")             # 3.0
print(f"‖a‖ = {np.linalg.norm(a):.2f}")        # 5.0
print(f"cos(θ) = {np.dot(a,b)/(np.linalg.norm(a)*np.linalg.norm(b)):.3f}")  # 0.6

# Cosine similarity (NLP/recommendation)
def cosine_sim(u, v):
    return np.dot(u, v) / (np.linalg.norm(u) * np.linalg.norm(v))

# ── Matrix operations ─────────────────────────────────────────────────────────
A = np.array([[2., 1.],
              [0., 3.]])

B = np.array([[1., 0.],
              [2., 1.]])

print("A @ B =")
print(A @ B)                     # matrix multiply (composition)
print(f"det(A) = {np.linalg.det(A):.2f}")   # 6.0 — volume scaling
print(f"rank(A) = {np.linalg.matrix_rank(A)}")   # 2 — full rank

A_inv = np.linalg.inv(A)
print("A @ A_inv ≈ I:", np.allclose(A @ A_inv, np.eye(2)))

# ── Eigendecomposition ────────────────────────────────────────────────────────
eigenvalues, eigenvectors = np.linalg.eig(A)
print(f"Eigenvalues: {eigenvalues}")          # [2. 3.]
print(f"Eigenvectors (columns):\n{eigenvectors.round(3)}")

# Verify: A @ v = λ * v
for i in range(len(eigenvalues)):
    v = eigenvectors[:, i]
    lam = eigenvalues[i]
    print(f"λ{i+1}={lam:.2f}, A@v = {A@v.round(3)}, λ*v = {(lam*v).round(3)}")

# Reconstruct A from eigendecomposition
Q = eigenvectors
Lambda = np.diag(eigenvalues)
A_reconstructed = Q @ Lambda @ np.linalg.inv(Q)
print("Reconstruction error:", np.linalg.norm(A - A_reconstructed))

# ── SVD ───────────────────────────────────────────────────────────────────────
M = np.random.randn(4, 3)               # 4×3 rectangular matrix
U, S, Vt = np.linalg.svd(M, full_matrices=False)
print(f"U: {U.shape}, S: {S.shape}, Vt: {Vt.shape}")

# Low-rank approximation (keep top-k singular values)
k = 2
M_approx = U[:, :k] @ np.diag(S[:k]) @ Vt[:k, :]
print(f"Rank-{k} approx error: {np.linalg.norm(M - M_approx):.4f}")

# ── PCA from scratch ─────────────────────────────────────────────────────────
X = np.random.randn(200, 5)
X -= X.mean(axis=0)                     # center
C = (X.T @ X) / (len(X) - 1)           # covariance matrix
eigenvalues, eigenvectors = np.linalg.eigh(C)   # eigh for symmetric matrices
idx = np.argsort(eigenvalues)[::-1]     # sort descending
PC = eigenvectors[:, idx[:2]]           # top-2 principal components
X_proj = X @ PC                         # project to 2D
print(f"Explained variance: {eigenvalues[idx[:2]] / eigenvalues.sum() * 100}")