#4. Matrix and vectors

Dot product (vectors)

import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
np.dot(a, b)  # 1*4 + 2*5 + 3*6 = 32
a @ b         # same, cleaner syntax

Matrix multiplication

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
A @ B
# [[19, 22],
#  [43, 50]]

Element-wise operations

A * B   # element-wise multiply (NOT matrix mul)
A + B   # element-wise add
A ** 2  # square every element

Transpose

A.T
# [[1, 3],
#  [2, 4]]

Inverse

np.linalg.inv(A)
# [[-2. ,  1. ],
#  [ 1.5, -0.5]]

Determinant

np.linalg.det(A)  # -2.0

Eigenvalues & eigenvectors

vals, vecs = np.linalg.eig(A)
# vals: [-0.37, 5.37]
# vecs: corresponding eigenvectors as columns

Solve linear system Ax = b

A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])
x = np.linalg.solve(A, b)  # [2. 3.]
# verify: A @ x == b

Norms

v = np.array([3, 4])
np.linalg.norm(v)        # L2 norm = 5.0
np.linalg.norm(v, ord=1) # L1 norm = 7.0
np.linalg.norm(A, 'fro') # Frobenius norm for matrix

Cosine similarity

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
cos_sim = np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))
# 0.9746 — used everywhere in NLP & recommendations

SVD (Singular Value Decomposition)

U, S, Vt = np.linalg.svd(A)
# U  — left singular vectors
# S  — singular values (diagonal)
# Vt — right singular vectors transposed
# used in PCA, dimensionality reduction, recommender systems

Broadcasting (no loops needed)

A = np.array([[1, 2, 3], [4, 5, 6]])
v = np.array([10, 20, 30])
A + v
# [[11, 22, 33],
#  [14, 25, 36]]
# numpy stretches v across every row automatically

Stacking vectors into a matrix

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
np.vstack([a, b])  # [[1,2,3],[4,5,6]] — row stack
np.hstack([a, b])  # [1,2,3,4,5,6]    — flatten & join
np.column_stack([a, b])  # [[1,4],[2,5],[3,6]] — col stack