lineaarialgebran

Lineaarialgebran, here understood as linear algebra, is a branch of mathematics that studies vector spaces and linear mappings between them. It provides a concise framework for describing lines, planes, and higher-dimensional analogues, and for solving systems of linear equations. The subject underpins much of science, engineering, computer science, economics, and data analysis, where relationships between quantities can be modeled as linear combinations.

The basic objects are vectors and matrices. A vector is an element of a vector space; matrices encode linear transformations relative to a basis. Key operations include vector addition, scalar multiplication, and matrix multiplication. Central notions include subspaces, bases, and dimension; linear independence; the rank of a matrix; and the notion of invertibility. Every linear map can be represented by a matrix, and questions about the map reduce to algebraic manipulations of this matrix.

Fundamental results include Gaussian elimination for solving linear systems, determinants characterizing invertibility, and eigenvalues and eigenvectors describing invariant directions under a transformation. More advanced topics cover diagonalization, canonical forms, and decompositions such as the singular value decomposition. These tools facilitate tasks such as solving systems efficiently, understanding geometric structure, and performing data reduction and least-squares fitting.

Applications span computer graphics, physics, optimization, machine learning, statistics, economics, and network analysis. Historical development began in the 19th century with works of Gauss and Cayley and matured through subsequent contributions from mathematicians such as Jordan, Weyl, and Halmos. Modern linear algebra emphasizes both theoretical foundations and computational methods, with widespread use in algorithms, simulations, and data-driven modeling.