📘 Chapter 11: Linear Algebra with NumPy — Powering Data Science and Machine Learning

🧠 “In the heart of every machine learning model lies a matrix. And NumPy makes that matrix manageable.”

Welcome to Chapter 11 of your NumPy learning journey!

So far, you’ve worked with arrays, filtered, reshaped, sorted, and simulated randomness. Now we’re diving into the brain of scientific computing: Linear Algebra — the foundation for:

• Machine learning algorithms

• Physics simulations

• Engineering calculations

• Data transformations

• Neural networks

Luckily, NumPy provides a powerful linear algebra module called np.linalg, which helps you work with matrices, equations, and decompositions.

🎯 What You’ll Learn:

• Matrix multiplication using dot() and matmul()

• Finding the inverse, transpose, and determinant

• Calculating eigenvalues and eigenvectors

• Solving systems of linear equations

Let’s turn numbers into vectors and vectors into insight. 🔍

🧱 1. Matrix Multiplication with dot() and matmul()

Let’s start with the basics.

🔹 Create Two Matrices:

import numpy as np

A = np.array([[1, 2],
              [3, 4]])

B = np.array([[5, 6],
              [7, 8]])

🔸 Using np.dot():

result = np.dot(A, B)
print("Dot Product:\n", result)

Output:

[[19 22]
 [43 50]]

🔸 Using np.matmul() (or @ operator):

result = np.matmul(A, B)
# or simply:
result = A @ B

🔄 2. Matrix Transpose

To swap rows and columns:

print("Transpose of A:\n", A.T)

Output:

[[1 3]
 [2 4]]

Useful in:

• Gram matrices

• Inner product checks

• Data alignment

🔁 3. Matrix Inverse with np.linalg.inv()

Only square matrices with non-zero determinant can be inverted.

inv_A = np.linalg.inv(A)
print("Inverse of A:\n", inv_A)

Output:

[[-2.   1. ]
 [ 1.5 -0.5]]

Check by multiplying:

print(np.dot(A, inv_A))  # Should return identity matrix

📏 4. Determinant with np.linalg.det()

The determinant tells you if a matrix is invertible.

det_A = np.linalg.det(A)
print("Determinant of A:", det_A)

Output:

-2.0000000000000004

If det == 0, the matrix is singular (non-invertible).

📈 5. Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are used in:

• Principal Component Analysis (PCA)

• Quantum mechanics

• Structural engineering

• System stability

eigvals, eigvecs = np.linalg.eig(A)

print("Eigenvalues:", eigvals)
print("Eigenvectors:\n", eigvecs)

You get:

• eigvals: scalar values (λ) such that Ax = λx

• eigvecs: vectors that remain in the same direction when transformed by A

🧠 6. Solving Systems of Linear Equations

Let’s say we have the following system:

2x + 3y = 8  
3x + 4y = 11

Convert to matrix form:

AX = B

Where:

A = np.array([[2, 3],
              [3, 4]])

B = np.array([8, 11])

Solve with np.linalg.solve():

solution = np.linalg.solve(A, B)
print("Solution [x, y]:", solution)

Output:

[1. 2.]

📦 Bonus: Pseudo-Inverse with np.linalg.pinv()

Used when the matrix is not square or not invertible.

M = np.array([[1, 2, 3], [4, 5, 6]])
pseudo_inv = np.linalg.pinv(M)
print("Pseudo-Inverse:\n", pseudo_inv)

Useful in:

• Linear regression

• Over-determined systems (more equations than unknowns)

🔍 Real-Life Applications of Linear Algebra in NumPy

⚠️ Common Mistakes to Avoid

🧾 Summary Table: Linear Algebra Essentials

🔚 Wrapping Up Chapter 11

Bravo! 🥳 You’ve entered the world of linear algebra, where numbers gain structure and logic becomes geometry.

With these tools, you can:

• Build the mathematical backbone of ML models

• Analyze real-world phenomena

• Solve complex systems

• Transform and compress data

From vector spaces to PCA, linear algebra in NumPy is your ticket to professional-level data science.

🧭 What’s Next?

Now that you’ve completed the NumPy Foundations Series, your next adventures might include:

• Pandas for data manipulation

• Matplotlib / Seaborn for visualization

• SciPy for scientific computing

• scikit-learn for machine learning