Linear Algebra with NumPy

Why linear algebra is useful

Linear algebra appears in:

• Regression
• Optimization
• Dimensionality reduction
• Correlation and covariance

NumPy provides numpy.linalg for common operations like decompositions, inverses, and determinants.

Dot product

dot

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

Matrix multiplication

* on two 2D arrays is element-wise. Matrix multiplication needs @ or np.matmul (equivalently np.dot for 2D arrays).

matmul

import numpy as np
 
A = np.array([[1, 2], [3, 4]])
B = np.array([[10, 20], [30, 40]])
 
print(A @ B)
print(np.matmul(A, B))

diagram Linear algebra toolbox mermaid

  flowchart LR
  A["Two matrices"] --> B["@ / np.matmul
matrix multiplication"]
  A --> C["np.linalg.solve(A, b)
solve Ax = b"]
  A --> D["np.linalg.inv(A)
matrix inverse"]
  A --> E["np.linalg.det(A)
determinant"]
  A --> F["np.linalg.eig(A)
eigenvalues/eigenvectors"]
  A --> G["np.linalg.norm(v)
vector length"]

numpy.linalg covers the standard matrix operations you need for regression, PCA, and solving systems.

Transpose

transpose

import numpy as np
 
A = np.array([[1, 2], [3, 4]])
print(A.T)

Determinant and inverse

det-inv

import numpy as np
 
A = np.array([[1, 2], [3, 4]])
 
det = np.linalg.det(A)
inv = np.linalg.inv(A)
 
print("det:", det)
print("inv:\n", inv)

Caution

Inverses can be numerically unstable. Often you should solve systems directly (np.linalg.solve).

Solve a system of equations

Solving A x = b directly with np.linalg.solve is faster and more numerically stable than computing inv(A) @ b.

solve

import numpy as np
 
A = np.array([[2, 1], [1, 3]])
b = np.array([8, 13])
 
x = np.linalg.solve(A, b)
print(x)

From the book

McKinney points out that X.T @ X (a matrix times its own transpose) is one of the most common patterns you’ll see in linear algebra code — it shows up constantly in regression and statistics, because it always produces a square, symmetric matrix that you can safely invert or decompose.

Eigenvalues and eigenvectors

eig

import numpy as np
 
A = np.array([[2, 0], [0, 3]])
vals, vecs = np.linalg.eig(A)
 
print("eigenvalues:", vals)
print("eigenvectors:\n", vecs)

Norms (vector length)

norm

import numpy as np
 
v = np.array([3, 4])
print(np.linalg.norm(v))  # 5

Next

Continue to: Statistical Functions in NumPy for mean/median/std/percentiles and basic descriptive analytics.

🧪 Try It Yourself

Exercise 1 – Dot Product

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Exercise 2 – Matrix Multiplication with @

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Exercise 3 – Solve a Linear System

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