How is NumPy linear algebra used in data science applications?

Linear algebra underpins almost all of machine learning — from computing gradients to PCA to solving systems of equations. NumPy's linalg submodule provides production-grade implementations of the core operations.

import numpy as np

# --- Solving a system of linear equations: Ax = b ---
# 2x + y = 8
# x + 3y = 11
A = np.array([[2, 1], [1, 3]])
b = np.array([8, 11])
x = np.linalg.solve(A, b)
print(x)   # [2.6  2.8]  — verify: A @ x ≈ b

# --- Matrix decompositions ---
M = np.array([[3, 1], [1, 3]], dtype=float)

# Eigenvalue decomposition
eigenvalues, eigenvectors = np.linalg.eig(M)
# eigenvalues = [4. 2.], eigenvectors (columns) = principal directions

# Singular Value Decomposition — used in PCA, recommendation systems
X = np.random.default_rng(42).random((100, 5))   # 100 samples, 5 features
X -= X.mean(axis=0)                               # centre
U, S, Vt = np.linalg.svd(X, full_matrices=False)
# S = singular values (square roots of eigenvalues of X^T X)
# Vt rows = principal components
# Project onto first 2 components:
X_pca = X @ Vt[:2].T    # shape (100, 2)

# --- Norms ---
v = np.array([3.0, 4.0])
np.linalg.norm(v)        # 5.0 — L2 norm
np.linalg.norm(v, ord=1) # 7.0 — L1 norm

# --- Matrix rank, determinant, inverse ---
np.linalg.matrix_rank(A)
np.linalg.det(A)
np.linalg.inv(A)   # only for square non-singular matrices
np.linalg.pinv(A)  # Moore-Penrose pseudoinverse for non-square

SVD is the engine behind PCA: the right singular vectors (rows of Vt) are the principal components, and the singular values tell you how much variance each component explains. Using full_matrices=False (economy SVD) is essential for tall matrices — it skips computing the large, unused portions of U.

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