الجبر الخطي لتعلم الآلة

import numpy as np

class="tok-comment"># ── Vectors and dot products ──────────────────────────────────────────────────
a = np.array([class="tok-num">3., class="tok-num">4.])
b = np.array([class="tok-num">1., class="tok-num">0.])

print(fclass="tok-str">"a·b = {np.dot(a, b):.2f}")             class="tok-comment"># class="tok-num">3.0
print(fclass="tok-str">"‖a‖ = {np.linalg.norm(a):.2f}")        class="tok-comment"># class="tok-num">5.0
print(fclass="tok-str">"cos(θ) = {np.dot(a,b)/(np.linalg.norm(a)*np.linalg.norm(b)):.3f}")  class="tok-comment"># class="tok-num">0.6

class="tok-comment"># Cosine similarity (NLP/recommendation)
def cosine_sim(u, v):
    return np.dot(u, v) / (np.linalg.norm(u) * np.linalg.norm(v))

class="tok-comment"># ── Matrix operations ─────────────────────────────────────────────────────────
A = np.array([[class="tok-num">2., class="tok-num">1.],
              [class="tok-num">0., class="tok-num">3.]])

B = np.array([[class="tok-num">1., class="tok-num">0.],
              [class="tok-num">2., class="tok-num">1.]])

print(class="tok-str">"A @ B =")
print(A @ B)                     class="tok-comment"># matrix multiply (composition)
print(fclass="tok-str">"det(A) = {np.linalg.det(A):.2f}")   class="tok-comment"># class="tok-num">6.0 — volume scaling
print(fclass="tok-str">"rank(A) = {np.linalg.matrix_rank(A)}")   class="tok-comment"># class="tok-num">2 — full rank

A_inv = np.linalg.inv(A)
print(class="tok-str">"A @ A_inv ≈ I:", np.allclose(A @ A_inv, np.eye(class="tok-num">2)))

class="tok-comment"># ── Eigendecomposition ────────────────────────────────────────────────────────
eigenvalues, eigenvectors = np.linalg.eig(A)
print(fclass="tok-str">"Eigenvalues: {eigenvalues}")          class="tok-comment"># [class="tok-num">2. class="tok-num">3.]
print(fclass="tok-str">"Eigenvectors (columns):\n{eigenvectors.round(class="tok-num">3)}")

class="tok-comment"># Verify: A @ v = λ * v
for i in range(len(eigenvalues)):
    v = eigenvectors[:, i]
    lam = eigenvalues[i]
    print(fclass="tok-str">"λ{i+class="tok-num">1}={lam:.2f}, Aclass="tok-dec">@v = {Aclass="tok-dec">@v.round(class="tok-num">3)}, λ*v = {(lam*v).round(class="tok-num">3)}")

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

class="tok-comment"># ── SVD ───────────────────────────────────────────────────────────────────────
M = np.random.randn(class="tok-num">4, class="tok-num">3)               class="tok-comment"># class="tok-num">4×class="tok-num">3 rectangular matrix
U, S, Vt = np.linalg.svd(M, full_matrices=False)
print(fclass="tok-str">"U: {U.shape}, S: {S.shape}, Vt: {Vt.shape}")

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

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