Concept Taxonomy
This document defines the categorical taxonomy for organizing the 300 concepts in the Applied Linear Algebra for AI and Machine Learning course.
Taxonomy Categories
| TaxonomyID | Category Name | Description |
|---|---|---|
| FOUND | Foundation Concepts | Basic mathematical building blocks including scalars, vectors, and fundamental operations that form the prerequisite knowledge |
| MATOP | Matrix Operations | Core matrix concepts, operations, and special matrix types essential for linear algebra computations |
| LINSYS | Linear Systems | Concepts related to systems of linear equations, solution methods, and matrix equation forms |
| TRANS | Transformations | Linear transformations, geometric operations (rotation, scaling, shear), and related structural concepts |
| DETERM | Determinants | Determinant computation, properties, and geometric interpretations |
| EIGEN | Eigentheory | Eigenvalues, eigenvectors, eigenspaces, and diagonalization concepts |
| DECOMP | Decompositions | Matrix factorization methods including LU, QR, Cholesky, and SVD |
| INPROD | Inner Products | Inner product spaces, orthogonality, projections, and related abstract vector space concepts |
| MLBASE | ML Foundations | Core machine learning concepts including data representation, PCA, regression, and gradient methods |
| NEURAL | Neural Networks | Deep learning concepts including neurons, layers, activation functions, and backpropagation |
| GENAI | Generative AI | Embeddings, attention mechanisms, transformers, and large language model concepts |
| OPTIM | Optimization | Optimization algorithms and methods for training machine learning models |
| IMGPROC | Image Processing | Computer vision concepts including image representation, convolution, and filtering |
| GEOM3D | 3D Geometry | Three-dimensional geometry, rotations, coordinate systems, and camera models |
| AUTON | Autonomous Systems | Sensor fusion, state estimation, SLAM, and autonomous navigation concepts |
Category Descriptions
FOUND - Foundation Concepts
The fundamental building blocks of linear algebra. These concepts are prerequisites for nearly everything else in the course, including scalars, vectors, vector operations, norms, and basic vector space theory.
MATOP - Matrix Operations
Essential matrix concepts and operations. Covers matrix notation, types of matrices (diagonal, triangular, symmetric, orthogonal), and core operations like multiplication, transpose, and inverse.
LINSYS - Linear Systems
Methods for representing and solving systems of linear equations. Includes Gaussian elimination, row operations, echelon forms, and solution analysis.
TRANS - Transformations
How matrices represent geometric transformations. Covers rotation, scaling, shearing, projection, and abstract concepts like kernel, range, and change of basis.
DETERM - Determinants
Determinant theory and applications. Includes computation methods, geometric interpretation as volume scaling, and applications like Cramer's rule.
EIGEN - Eigentheory
The study of eigenvalues and eigenvectors - one of the most important topics in applied linear algebra. Covers characteristic polynomials, diagonalization, spectral theorem, and power iteration.
DECOMP - Decompositions
Matrix factorization techniques. Each decomposition has specific use cases: LU for solving systems, QR for least squares, Cholesky for symmetric positive definite matrices, and SVD for general applications.
INPROD - Inner Products
Abstract theory of inner product spaces. Covers orthogonality, Gram-Schmidt process, projections, least squares, and the four fundamental subspaces.
MLBASE - ML Foundations
Core machine learning concepts that rely on linear algebra. Includes data representation, covariance analysis, PCA, linear regression, regularization, and gradient descent.
NEURAL - Neural Networks
Deep learning architecture and computation. Covers the linear algebra of neural networks including weight matrices, forward propagation, backpropagation, and specialized layers.
GENAI - Generative AI
Modern generative AI concepts. Focuses on the linear algebra behind transformers, attention mechanisms, embeddings, and large language models.
OPTIM - Optimization
Optimization algorithms for training. Covers gradient-based methods, second-order optimization, and constrained optimization techniques.
IMGPROC - Image Processing
Computer vision fundamentals. Covers image representation, convolution, filtering, frequency domain analysis, and feature detection.
GEOM3D - 3D Geometry
Three-dimensional geometric concepts. Includes coordinate systems, rotation representations (Euler angles, quaternions), camera models, and stereo vision.
AUTON - Autonomous Systems
Sensor fusion and autonomous navigation. Covers Kalman filtering, SLAM, localization, object tracking, and path planning.