LLM & Transformer Mathematics: AI Explained Step by Step

Large Language Models have transformed artificial intelligence, but using an LLM and understanding how one actually works are two very different things. This course takes you beneath the surface of modern AI and explains the core mathematics behind Transformers and Large Language Models step by step. Rather than treating concepts such as embeddings, positional encoding, Query-Key-Value matrices and attention as black boxes, we break them down visually and mathematically so you can understand what is actually happening inside a Transformer.

You will begin with the foundations of representing language numerically. We explore categorical encoding, tokenization, vocabularies and token encoding before moving into embeddings and continuous vector spaces. From there, we tackle one of the most important concepts in Transformer architecture: positional encoding.

You will see why Transformers need positional information, how sine and cosine functions are used to represent token positions, why even and odd dimensions are treated differently, and how the underlying equations work. The course then moves into the heart of the Transformer: the attention mechanism. Instead of simply showing you an attention formula, we work through the process step by step.

You will explore Query, Key and Value matrices, calculate vectors, examine dot products and vector alignment, calculate raw attention scores and visualize what those scores represent. We then convert those raw scores into attention weights using softmax and normalization before combining them with Value vectors to produce contextual representations. Finally, you will explore multi-head attention and see how multiple attention heads allow a Transformer to capture different relationships within the same sequence.

What you will exploreThroughout the course, you will work with concepts including:• Tokenization and vocabularies• Numerical token representations• Embeddings and vector spaces• BERT and DistilBERT concepts• Positional encoding• Sine and cosine functions• Query, Key and Value matrices• Dot products and vector alignment• Raw attention scores• Softmax and normalization• Attention weights• Context vectors• Self-attention• Multi-head attention• Masked language modelingA mathematical approach without unnecessary complexityThe goal of this course is not to overwhelm you with abstract equations. Instead, mathematical concepts are introduced when they are needed and connected directly to the Transformer components they explain. You will follow calculations step by step and develop intuition for why the mathematics works, not simply memorize formulas.

This makes the course suitable for learners who already work with AI as well as those building a stronger technical foundation in Large Language Models. Why take this course? Many AI courses focus on using existing LLMs, calling APIs or building applications around models.

Those are valuable skills, but they don’t necessarily explain what happens inside the model. This course focuses on that missing layer. By the end, terms such as embeddings, positional encoding, Query-Key-Value, attention scores, softmax, context vectors and multi-head attention should no longer feel like abstract Transformer terminology.

You will be able to connect these concepts mathematically and understand how they work together to transform tokens into context-aware representations. If you want to move beyond simply using Large Language Models and develop a deeper understanding of the mechanisms that make Transformers work, this course is designed for you.