ruumialasid

Ruumialasid refers to the concept of a basis of a vector space in Estonian mathematical terminology. It denotes a set of vectors that both spans the space and is linearly independent. The plural form is ruumialasid.

In a finite-dimensional vector space V over a field F, a set B = {b1,...,bn} is a ruumialus if every v in V can be written uniquely as v = a1 b1 + ... + an bn with scalars ai in F, and no nontrivial linear combination yields the zero vector. Consequently, n equals the dimension of V.

The standard basis in R^n, consisting of the vectors with a single 1 and zeros elsewhere, is the canonical example. In function spaces, common bases include {1, x, x^2, ..., x^n}. The choice of basis is not unique; many bases can span the same space.

Practical methods to identify or construct a basis include Gaussian elimination on a generating set to find pivot columns, or starting from a linearly independent set and extending it. In inner product spaces, bases may be chosen to be orthonormal via Gram–Schmidt, which simplifies coordinates and projections.

For infinite-dimensional spaces, one distinguishes between Hamel bases (algebraic basis) and Schauder bases (topological basis for convergent series). The concept of a ruumialas is foundational in linear algebra, geometry, physics, and computer science, where coordinates relative to a basis enable simple representation and computation.