Linearalgebra

Linear algebra is the branch of mathematics that studies vectors, vector spaces, and linear mappings between them. It emphasizes linear relations and operations that preserve addition and scalar multiplication, enabling the analysis of systems of linear equations, geometric transformations, and the structure of spaces.

Core objects include vectors and matrices. A vector is an element of a vector space; a matrix represents a linear map relative to chosen bases. Finite- and infinite-dimensional spaces are treated, and key concepts such as linear independence, spanning sets, bases, and dimension describe their structure.

Matrix theory connects algebra with geometry. For a linear map between finite-dimensional spaces, a matrix encodes its action in a basis. Important properties include rank, determinant, and invertibility, which determine solvability of linear systems and transform behavior. Eigenvalues and eigenvectors reveal invariant directions and facilitate diagonalization or Jordan forms. Inner products define lengths and angles, while orthogonality and orthonormal bases, via Gram–Schmidt, support projections and least-squares methods.

Applications span computer graphics, physics, engineering, statistics, and data science. Linear algebra provides the foundations for algorithms in machine learning, signal processing, numerical analysis, and optimization, as well as methods for data representation, dimensionality reduction, and solving large-scale systems.

History and notation: ideas emerged in the 19th century from Gauss, Grassmann, and others, with substantial development in the 20th century as a formal theory of vector spaces and matrices. Today linear algebra is a unifying framework across science and engineering.