Linear Algebra

The study of vector spaces, matrix operations, and linear transformations between them.

• core objects:: vectors, matrices, tensors
• eigenvalues and eigenvectors reveal invariant directions under transformation
• determinant measures volume scaling; rank measures dimensional span
• The spectral theorem decomposes symmetric matrices into orthogonal eigenbases
• Foundation of machine learning, quantum mechanics, signal processing, and optimization
• singular value decomposition generalizes eigendecomposition to rectangular matrices
• inner product defines angles and distances, enabling geometry in arbitrary dimensions
• Related: calculus, statistics, fourier transform, differential equations, category theory