koordinatvektor

A koordinatvektor, or coordinate vector, is a representation of a vector in a specific coordinate system. It lists the scalar components of the vector along the basis vectors of that system. For example, in a two-dimensional Cartesian coordinate system with standard basis vectors i = (1, 0) and j = (0, 1), a vector v can be written as v = ax + by, where a and b are scalars. The coordinate vector of v in this system is then represented as [a, b] or (a, b).

The concept of a coordinate vector is fundamental in linear algebra and geometry. The choice of basis vectors significantly influences the appearance of the coordinate vector, even though it represents the same underlying geometric object. Different coordinate systems, such as polar coordinates or curvilinear coordinates, will result in different coordinate vectors for the same physical vector.

When working with vectors, it is crucial to be aware of the coordinate system being used. Operations like vector addition and scalar multiplication are performed component-wise on the coordinate vectors. For instance, if u = [u1, u2] and v = [v1, v2], then u + v = [u1 + v1, u2 + v2]. The transformation of coordinate vectors between different bases is a key topic in advanced linear algebra, often involving change-of-basis matrices.