Graphics and Data
Linear algebra powers computing: data and geometry are vectors and matrices, and computation is matrix arithmetic.
In computer graphics, 3D scenes are point vectors. Linear transformations (matrix multiplications) rotate, scale, and project them. The GPU is a machine for massive matrix-vector products.
In machine learning, datasets are matrices (rows are samples, columns are features).
| Tool | Definition | Computing Use |
|---|---|---|
| PCA | Top eigenvectors of covariance matrix | Compresses data along informative directions |
| Cosine similarity | Normalized dot product | Measures similarity of feature vectors or word embeddings |
| Neural network | Matrix multiply plus nonlinearity | Weights times inputs; training adjusts matrices |
Networks and the Big Picture
In networks and ranking, a graph's connections form an adjacency matrix. PageRank computes the dominant eigenvector of the web's link matrix to rank pages, a technique used across social and citation networks.
Graphics pipeline formula: where is a vector, and are transformation matrices.
Common pitfall: Treating these applications as unrelated tricks. Graphics, PCA, neural networks, and PageRank are all the same core operations: matrix-vector products, dot products, and eigenvectors applied to different data.