Lineaaralgebras

Lineaaralgebras is a branch of mathematics that studies vectors, vector spaces, and linear transformations between them. It focuses on linear relationships, and uses algebraic methods to analyze geometric and numerical problems. The subject is defined over a field, commonly the real or complex numbers, and can be developed for finite or infinite-dimensional spaces.

Key objects include vectors and subspaces, bases and dimension, and linear maps. A vector space has operations of addition and scalar multiplication; a basis is a set of vectors that span the space and are linearly independent; the number of basis vectors is the dimension. Linear maps preserve addition and scalar multiplication, and are often represented by matrices with respect to chosen bases.

Matrix methods underpin computation in lineaaralgebras. Systems of linear equations are analyzed by row reduction, yielding the rank, the solution space (null space), and the image (column space). Matrix multiplication, inverses, and determinants (where defined) provide tools for solving and understanding transforms.

Eigenvalues and eigenvectors capture intrinsic directions of linear transformations; diagonalization and the spectral theorem describe when a transformation can be simplified. Inner product spaces introduce geometry: dot products, norms, orthogonality, and projections. The Gram–Schmidt process builds orthonormal bases, enabling stable computations.

Applications range from computer graphics and machine learning to physics, engineering, and economics. Linear algebra provides the language for systems modeling, data decomposition, and optimization. The subject developed in the 19th century with contributions from Gauss, Cayley, and others, and continues to be foundational in mathematics and applied sciences.