Lineaarialgebraan

Lineaarialgebraan, commonly spelled lineaire algebra or linear algebra, is the branch of mathematics that studies vector spaces and linear mappings between them, as well as the structures that arise from them, such as matrices. It provides a framework for analyzing systems of linear relations and transformations in various dimensions.

Core objects include vectors and scalars, vector spaces formed over a field, and linear combinations. Key concepts are span, linear independence, bases, dimension, and subspaces. These ideas establish coordinate systems and enable the description of geometric and algebraic properties in a compact form.

Linear transformations and matrices are central to the subject. A linear map between vector spaces preserves addition and scalar multiplication, and every finite-dimensional linear map can be represented by a matrix relative to chosen bases. Matrix operations correspond to composition of linear maps and enable efficient computation and analysis.

Important topics encompass matrices, determinants, and invertibility, as well as row reduction and Gaussian elimination for solving systems of linear equations. Eigenvalues and eigenvectors reveal intrinsic directions and scales under linear maps, with diagonalization and Jordan forms providing canonical representations. Inner product spaces, norms, and orthogonality lead to projections and orthonormal bases, obtainable via Gram–Schmidt.

Applications and methods span numerous fields. Linear algebra underpins solving linear systems, least squares approximations, and matrix decompositions such as LU, QR, and Singular Value Decomposition. It is foundational in computer graphics, physics, engineering, statistics, data science, and machine learning, where high-dimensional data and linear models are routinely analyzed.