Baasõlist

Baasõlist is a term used in Estonian mathematics and computer science to denote an ordered collection of elements that forms a basis for a vector space, module, or other algebraic structure. The word combines baas, meaning base or basis, with a suffix indicating a list, underscoring its nature as an ordered sequence.

In a vector space V over a field F, a baasõlist consists of vectors v1, v2, ..., vn that are linearly independent and span V. The dimension of V equals n. Every vector v in V can be written uniquely as v = a1 v1 + ... + an vn, and the n-tuple (a1, ..., an) is the coordinate vector of v with respect to the baasõlist. The matrix whose columns are the basis vectors is called the basis matrix and enables a linear transformation between coordinate representations and standard representations.

The concept generalizes to modules over a ring and to bases in inner product spaces, where additional properties such as orthogonality or orthonormality may be considered. A change of baasõlist corresponds to a basis change, computable via a transition matrix, which converts coordinates from one basis to another.

Construction and computation commonly involve methods like Gaussian elimination or row reduction to extract a maximal linearly independent set from a given set of generators. In applications, baasõlist simplifies the representation of vectors and linear transformations, facilitates coordinate computations, and underpins algorithms for solving linear systems.

Examples: In R^3, the standard baasõlist is [(1,0,0), (0,1,0), (0,0,1)]. In data modeling, a baasõlist can metaphorically refer to a base set of features from which others are derived through linear combinations.

See also: Basis, Linear independence, Span, Dimension, Basis change, Dual basis.