2 "Linear Algebra" Posts

How Large Language Models (LLMs) Think: Turning Meaning into Math

Exploring how large language models use linear algebra to create geometric meaning

Oct 7, 2025 • Learning Paths: How Large Language Models Work · Inside the Transformer • Category: AI and the Mathematics of Language • Tags: #AI · #embeddings · #geometry · #linear algebra · #LLM

What You'll Learn

How an LLM turns words and tokens into numerical vectors it can process
How distance and direction in embedding space can represent relationships in meaning
Why linear algebra and geometry are two ways of describing the same internal structure
How matrix operations transform information as it moves through a model
Why high-dimensional spaces can represent many subtle features of language at once
How probability guides a model from its current context toward the next token

Using SymPy in Python When NumPy Isn't Enough

Choosing exact symbolic mathematics when floating-point approximations are not good enough

Sep 5, 2025 • Category: Python Techniques and Tooling • Tags: #linear algebra · #Python · #SymPy · #symbolic math

What You'll Learn

Why floating-point numbers cannot represent many ordinary decimal values exactly
How SymPy keeps rational values exact instead of introducing floating-point approximations
When tolerance checks such as math.isclose are appropriate and when exact math matters
How symbolic computation can calculate derivatives and solve equations without numerical approximation
When SymPy is a better choice than NumPy for a mathematical problem
How SymPy and NumPy complement each other as tools for precision and performance

Behind the Code

My focus is the engineering beneath modern AI systems. After several years of building applications with AI through APIs, my interests have moved below that abstraction layer: understanding how models are architected, trained, evaluated, and served, why they’re designed the way they are, and the engineering tradeoffs behind those decisions.

That direction follows a pattern across my forty-year career as a software developer. I’ve primarily built applications, but I’ve repeatedly found myself pushing beyond the application itself and building the systems, abstractions, and tooling behind it: metaprogramming layers, dynamic runtimes, configuration-driven engines, and automation platforms designed to outlive the original feature request.

The pattern shows across my career: Forms Express at AT&T, a four-layer dynamic runtime shipped for under $500K against a $2M consulting bid; VeriSign’s first-ever customer self-service mobile activation system, which won the 2003 SUPERQuest Award; and TAG at Microsoft, a production LLM orchestration platform that shipped nearly 100 Terraform articles by parsing XML templates, routing prompts to Azure OpenAI, and composing results with browser-driven Azure portal capture. Along the way: C++ MVP, nine technology books, 100+ articles for .NET Programming Tips & Techniques, and helping run CodeGuru, the largest Windows developer community of its era.

After 21 years at Microsoft, I’m now full-time on what’s next: completing my Master’s in Computer Science and publishing at Signal & Syntax, where I research and write about the mathematics and engineering underneath modern AI. I use mathematical models, experiments, and small implementations to explore everything from gradient descent and attention to inference-time behavior and training infrastructure; not just how to use these systems through an API, but how the machinery beneath that API actually works.

Connect with me on LinkedIn .