🔢 Introduction to Linear Algebra Using NumPy (for Beginners)

Linear algebra is the backbone of many fields in science, engineering, machine learning, and computer graphics. If you’re learning data science, AI, or numerical computing, understanding linear algebra — especially with NumPy in Python — is essential.

In this tutorial, you’ll learn:

What linear algebra is
Why it matters in programming
How to perform core operations (vectors, matrices, dot products, solving equations) using NumPy

Let’s get started!

📚 What is Linear Algebra?

Linear algebra is a branch of mathematics dealing with vectors, matrices, and linear transformations.
It helps model and solve systems of linear equations and perform calculations on multi-dimensional data.

Common objects:

Scalars – single values (e.g., 5)
Vectors – 1D arrays (e.g., [3,5,7][3, 5, 7][3,5,7])
Matrices – 2D arrays (e.g., 2×32 \times 32×3 tables of numbers)
Tensors – higher-dimensional generalizations (used in deep learning)

🧮 Why Use NumPy for Linear Algebra?

NumPy is a powerful Python library that provides fast, efficient tools for numerical computations, including linear algebra.

Benefits:

Optimized C backend = fast matrix operations
Easy syntax for mathematical expressions
Integrated with machine learning tools like scikit-learn and TensorFlow

🧑‍💻 Getting Started with NumPy

1. Import the library

import numpy as np

2. Creating Vectors and Matrices

# 1D vector
v = np.array([2, 4, 6])

# 2D matrix
M = np.array([[1, 2], [3, 4]])

print("Vector:", v)
print("Matrix:\n", M)

3. Matrix Addition and Scalar Multiplication

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Addition
C = A + B

# Scalar multiplication
D = 3 * A

print("A + B =\n", C)
print("3 * A =\n", D)

4. Matrix Multiplication (Dot Product)

# Dot product of two vectors
a = np.array([1, 2])
b = np.array([3, 4])

dot = np.dot(a, b)
print("Dot product:", dot)

# Matrix multiplication
A = np.array([[1, 2], [3, 4]])
B = np.array([[2, 0], [1, 2]])

product = np.matmul(A, B)
print("Matrix product:\n", product)

5. Transpose and Inverse

# Transpose
M = np.array([[1, 2], [3, 4]])
T = M.T

# Inverse
inv = np.linalg.inv(M)

print("Transpose:\n", T)
print("Inverse:\n", inv)

6. Solving Systems of Linear Equations

Solving Ax=b

A = np.array([[2, 1], [1, 3]])
b = np.array([8, 13])

x = np.linalg.solve(A, b)
print("Solution x:", x)

📈 Where Is Linear Algebra Used?

📊 Data Science – dimensionality reduction (PCA), regression
🧠 Machine Learning – neural networks, gradient descent
🎮 Computer Graphics – 3D transformations
📐 Engineering – simulations and control systems

🧩 Summary

In this introduction, you’ve learned how to:

Create vectors and matrices in NumPy
Perform basic operations like dot product, transpose, inverse
Solve systems of equations with NumPy

Linear algebra isn’t just theoretical — with tools like NumPy, you can apply it directly to solve real problems in programming and data analysis.

learnmathcode.com