Vektorialgebran

Vektorialgebran is a branch of mathematics that studies vectors and the algebraic rules governing their combination. Vectors are objects that can be added together and multiplied by scalars, usually from a field such as the real numbers. The subject investigates properties of vector spaces, operations, and structures that arise from these rules, in spaces of any finite dimension and, in many theories, infinite dimensions.

Core concepts include vector addition, scalar multiplication, linear combinations, span, and linear independence. A basis is a minimal spanning set, and its size defines the space’s dimension. Linear transformations map vectors to vectors and are often represented by matrices relative to a chosen basis. Tools such as inner products (dot product) define length and angles, while in three dimensions the cross product yields a vector perpendicular to the plane of two vectors. More generally, inner product spaces and normed spaces support distance and projection operations.

Applications of vektorialgebran appear in physics, computer graphics, engineering, statistics, and data science. The theory underpins algorithms for solving linear systems, eigenvalue problems, and many numerical methods. The field overlaps with linear algebra and is referred to in many languages by terms closely related to vector algebra.