Understanding LLMs – A Mathematical Approach to the Engine Behind AI

Large Language Models (LLMs) are among the most transformative AI technologies of our time. They can generate fluent text, hold natural conversations, translate across languages, summarize complex documents, and even produce working code—tasks once thought to be uniquely human.

What powers these abilities isn’t magic—it’s mathematics. Concepts you may remember from high school or college—probability, matrices, vectors, and gradients— form the backbone of how LLMs “think.” By uncovering these foundations, we gain a clearer understanding of why these models work the way they do.

This book is written not only for engineers but also for data scientists, researchers, and anyone curious about AI. Rather than stopping at “how to use” LLMs, it guides you toward answering a deeper question: “Why do they work?”. With that knowledge, you’ll be able to apply AI more responsibly, more creatively, and with greater confidence.

For the complete version, see Understanding LLMs Through Math: The Inner Workings of Large Language Models (LLM Master Series) (available on Kindle and in print).

1. Mathematical Foundations for Understanding LLMs
   1. Getting Comfortable with Mathematical Notation
   2. Basics of Probability and Dialogue Generation
   3. Information Theory and Entropy
   4. Intuition for Linear Algebra and Vector Spaces
2. Core Concepts of LLMs
   1. What Is an LLM?
   2. Fundamental Components of LLMs
   3. Natural Language Processing (NLP) Overview
   4. Understanding Transformer Models
3. Mathematical Models Under the Hood
   1. Probability and Statistics for Language Generation
   2. Dialogue Modeling with Probability
4. The Mathematics of Transformers
   1. The Self-Attention Mechanism
   2. Multi-Head Attention Explained
5. Optimizing Models with Gradient Descent
   1. The Role of Loss Functions
   2. Gradient Descent and Backpropagation
6. Large Datasets and Training in Practice
   1. Data Preprocessing
   2. Mini-Batch Learning and Computational Efficiency
7. Practical Applications of LLMs
   1. Text Generation and Summarization
   2. Question Answering and Translation
8. Challenges and Future Outlook
   1. Model Size and Computational Costs
   2. Bias and Ethical Challenges
9. Key Considerations for Engineers
   1. Next Steps for Learning
   2. Resources for Implementation

Now, let’s dive into the first section: Understanding LLMs – A Mathematical Approach to the Engine Behind AI.