Learning Notes
A curated notebook of daily educational milestones, interactive sandboxes, and conceptual findings in physics, engineering, and mathematics.
Intuitive Linear Algebra: Vectors as Directions, Forces, and Data
Deconstructing the core protagonist of linear algebra—the vector—through geometric intuition, physical forces, and machine learning representation.
- What is a Vector? It's not just a list of numbers; it's a pointer in space. Think of it as a set of instructions: go 3 units east, 4 units north. Stacking them vertically (a column vector) is just our bookkeeping method.
- The Transpose (T): Stacking numbers vertically takes up a lot of paper. The transpose operation is just a space-saving rotation that tips a column vector onto its side into a row vector, written as (x, y, z)ᵀ to save vertical lines.
- Standard Basis / One-Hot Vectors: The primary directions—like pure East (e₁), pure North (e₂). In machine learning, these are 'one-hot' vectors, where a single '1' lights up to represent a category (like a light switch for a single option) while everything else is 0.
- Vector Addition: Geometrically, it's placing arrows head-to-tail. Physically, it's combining forces—if two people pull a box in different directions, the box moves along the diagonal of the parallelogram they form.
- Scalar Multiplication: Stretching, shrinking, or flipping. Multiplying by a positive scalar stretches the arrow; multiplying by a negative scalar reverses its direction (like running backwards).
- Vector Spaces: A playground with rules. Any combination of stretching and adding vectors stays inside the playground, governed by natural arithmetic laws like commutativity and distributivity.
Agentic Workflows & Next.js Routing Integration
Explored how to dynamically update Next.js applications and integrate custom sub-routes like `/learning` with gray-matter configurations.
Semi-Supervised Learning & Graph Laplacians
Researched total variation (TV) methods vs. graph Laplacians for classification tasks on high-dimensional datasets under Prof. Farid Bozorgnia.
CAD Fabrication Constraints & Strength Analysis
Studied fabrication tolerances and mechanical stress analysis of structural joints in CAD modeling.
WebMCP & AI Assistant Agentic Tool Calling
Researched Web Model Context Protocol (WebMCP) and declarative tool annotations for frontends.
Spectral Methods & Non-Local Operators
Explored non-local diffusion operators and fractional Laplacians in Euclidean space.