Think Linear Algebra, 1st edition

Think Linear Algebra is a code-first, case-based introduction to the most widely used concepts in linear algebra, designed for readers who want to understand and apply these ideas — not just learn them in the abstract. Each chapter centers on a real-world problem like modeling traffic in the web, simulating flocking birds, or analyzing electrical circuits. Using Python and powerful libraries like NumPy, SciPy, SymPy, and NetworkX, readers build working solutions that reveal how linear algebra provides elegant, general-purpose tools for thinking and doing.

This book is for readers who may have struggled with traditional math instruction, or who want a more intuitive, hands-on way to learn. By working in Jupyter notebooks, readers get instant feedback as they write code, run simulations, visualize results, and explore what-if scenarios. Rather than beginning with mathematical formalism, Think Linear Algebra starts with meaningful applications and builds up the theory when it's needed. The result is a uniquely practical and empowering introduction to linear algebra as a language for solving real problems.

Linear algebra is foundational for machine learning, scientific computing, and computer graphics — fields with enormous demand and growth. From search engines and GPS tracking to signal processing and structural engineering, linear algebra is the language behind many of the technologies that shape our world. This book shows you how to use it effectively in your own work.

Available Chapters

Here are the chapters that are available now. More coming soon!

Chapter 1: The Power of Linear Algebra

• Click here to run Chapter 1 on Colab: Introduces matrix multiplication and eigenvectors through a network-based model of museum traffic, and implements the PageRank algorithm for quantifying the quality of web pages.

Chapter 4: Projection

• Click here to run Chapter 4 on Colab: Presents vector projection, vector rejection, orthogonality and the dot product, using the elastic collision of pool (billiards) balls as an example.

Chapter 5: To Boldly Go

• Click here to run Chapter 5 on Colab: Uses matrices scale, rotate, shear, and translate vectors. Applies these methods to 2D compute graphics, including a reimplementation of the classic video game Asteroids.

Chapter 7: Systems of Equations

• Click here to run Chapter 7 on Colab: Applies LU decomposition and matrix equations to analyze electrical circuits. Shows how linear algebra solves real engineering problems.

Chapter 8: Null Space

• Click here to run Chapter 8 on Colab: Investigates chemical stoichiometry as a system with multiple valid solutions. Introduces concepts of rank and nullspace to describe the solution space.

Chapter 9: Truss In the System

• Click here to run Chapter 9 on Colab: Models structural systems where the unknowns are vector forces. Uses block matrices and rank analysis to compute internal stresses in trusses.

Chapter 10: Regression

• Click here to run Chapter 9 on Colab: Computes least squares regression using QR decomposition and orthogonality equations. Uses multiple regression and data from the General Social Survey (GSS) to explore the relationship of political ideology with time, age, and year of birth.

Working with the code

We recommend using Google Colab to run the notebooks — it's free, requires no setup, and works in your browser. Simply click the "Click here to run" links above for each chapter.

If you prefer to work locally, follow these steps:

1. Get the code (choose one option)

   Option A: Download zip file (simplest)

   • Download the zip file from GitHub
   • Extract the zip file and cd ThinkLinearAlgebra

   Option B: Clone with git

   git clone --depth 1 https://github.com/AllenDowney/ThinkLinearAlgebra.git
   cd ThinkLinearAlgebra

2. Set up your environment (choose one option)

   Option A: Conda environment (recommended)

   make create_environment
   conda activate ThinkLinearAlgebra
   make requirements

   Option B: Install packages individually

   pip install numpy scipy sympy networkx matplotlib jupyter pandas

3. Run the notebooks

   • With solutions: Open notebooks directly from the soln/ directory
   • Without solutions (for exercises): Use notebooks from the chapters/ directory, which have blank cells where you can work on exercises

Repository Structure

ThinkLinearAlgebra/
├── jb/                     # Jupyter Book source files
│   ├── index.md           # Landing page
│   ├── _config.yml        # Book configuration
│   ├── _toc.yml          # Table of contents
│   └── build.sh          # Build script for HTML version
├── chapters/              # Exercise notebooks (blank cells for student work)
│   ├── *.ipynb           # Chapter notebooks with exercises
│   └── build.sh          # Build script to create zip file
├── soln/                  # Solution notebooks and code
│   ├── *.ipynb           # Individual chapter notebooks with solutions
│   ├── utils.py          # Helper functions and utilities
│   └── *.py              # Additional Python modules
├── data/                  # Datasets used in the book
├── papers/               # Research papers and references
├── tests/                # Unit tests for utility functions
├── requirements.txt      # Python package dependencies
└── README.md            # This file

The main content is organized in three ways:

• soln/: Complete notebooks with solutions for reference
• chapters/: Exercise notebooks with blank cells for student work
• jb/: Source files for building the HTML version of the book

License

This book is available under a Creative Commons license, which means that you are free to copy, distribute, and modify it, as long as you attribute the source and don't use it for commercial purposes.