Linear Algebra is the branch of mathematics concerned with vector spaces, linear transformations and systems of linear equations. Its central objects include vectors, matrices, determinants, eigenvalues and eigenvectors, and operations such as matrix multiplication and inversion. The field provides the language for representing and solving problems in geometry, physics, computer graphics and data analysis. It is foundational to machine learning, where data, model parameters and transformations are expressed and manipulated as tensors.
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- At its core, linear algebra studies linear maps between vector spaces and how they can be represented by matrices once a basis is chosen. Solving a system of linear equations, computing the rank of a matrix and finding a basis for a subspace are recurring tasks that reduce to a small set of standard procedures such as Gaussian elimination.
- Eigenvalues and eigenvectors describe directions that a transformation merely scales, and the related spectral and singular value decompositions reveal structure in matrices that is essential to dimensionality reduction and numerical stability. These decompositions underpin techniques like principal component analysis and the solution of least-squares problems.
- In computing, linear algebra is the workhorse of graphics, optimisation and machine learning. Neural networks express their layers as matrix multiplications followed by nonlinearities, and efficient hardware for these operations is a major driver of progress in artificial intelligence.