Linear Algebra

Comprehensive guide to linear algebra, covering matrices, vectors, transformations, and applications in computer science and machine learning

Essential articles on linear algebra, the mathematical foundation for machine learning, computer graphics, and scientific computing.

Linear algebra is the mathematics of vector spaces and linear mappings between them. It provides the language and tools for representing and solving systems of linear equations, transforming geometric objects, and performing operations on high-dimensional data. At its core are vectors (ordered lists of numbers representing points in space), matrices (rectangular arrays representing linear transformations), and operations like matrix multiplication, which composes successive transformations. The concepts of linear independence, basis, span, and dimension describe the structure of vector spaces, while eigenvalues and eigenvectors reveal the intrinsic behavior of linear transformations — the directions that are stretched but not rotated.

In computer science, linear algebra is ubiquitous. Machine learning models (neural networks, PCA, SVMs, word embeddings) are fundamentally linear algebra operations: a forward pass through a neural network is repeated matrix-vector multiplications and nonlinear activations. Computer graphics uses 4x4 transformation matrices to rotate, scale, translate, and project 3D objects onto 2D screens. Google’s PageRank algorithm uses eigenvector computation to rank web pages. Data science relies on matrix factorizations (SVD, QR, Cholesky) for dimensionality reduction, recommendation systems, and solving least-squares problems. Understanding linear algebra gives developers intuition for how these algorithms work and how to debug them when they fail.

Why It Matters

Linear algebra is the mathematical foundation of modern computing — every neural network, graphics pipeline, search engine, and recommendation system relies on vector and matrix operations. Engineers who understand linear algebra concepts can design better algorithms, optimize performance, and debug ML models more effectively.

Core Topics

Fundamentals (1/1 Complete) ✅

  1. Matrix and Linear Transformation - Introduction to matrices and linear transformations