Building Large Language Models requires a solid grasp of the underlying mathematical concepts. While deep learning frameworks handle many low-level calculations, understanding the mathematics is necessary for designing effective architectures, interpreting training dynamics, debugging issues like exploding gradients ($\nabla L \to \infty$), and implementing custom components.

This chapter revisits key mathematical areas that form the foundation for the techniques discussed throughout the course. We will review:

• Linear Algebra: Focusing on vectors, matrices, and operations fundamental to representing data and transformations in neural networks.
• Calculus: Covering derivatives, gradients (e.g., $\nabla f(x)$), and the chain rule, which are essential for understanding backpropagation and optimization algorithms like gradient descent: $\theta_{t+1} = \theta_t - \eta \nabla_{\theta} J(\theta)$
• Probability and Statistics: Discussing concepts like probability distributions, entropy, and sampling, which are central to language modeling and evaluation.
• Numerical Stability: Briefly touching upon practical issues encountered when implementing deep learning models.

We will also establish the mathematical notation used in subsequent chapters to ensure clarity. This review aims to refresh these concepts, providing the necessary background to fully engage with the technical details of LLM construction and training.