basisvector

Basis vector

Basis vector is a term used in linear algebra to denote an element of a basis of a vector space. A basis is a set of vectors that are linearly independent and that span the space; a basis vector is any one member of that set. In Euclidean space R^n, the standard basis consists of the n vectors e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), ..., en = (0, ..., 0, 1). These vectors form a basis for R^n and constitute an orthonormal basis with respect to the usual inner product.

Any vector v in R^n can be written uniquely as a linear combination v = v1 e1 + v2 e2 + ... + vn en, where the scalars vi are the coordinates of v relative to the chosen basis. The basis determines a coordinate system: different bases provide different coordinate descriptions of the same vectors.

Bases are not unique. There are many possible sets of vectors that can serve as a basis for the same space, and the vectors in a basis are not required to be orthogonal or of unit length unless the basis is specifically chosen to be orthonormal (as with the standard basis in R^n). In finite-dimensional spaces, the number of basis vectors equals the dimension of the space.

The concept generalizes to any vector space over a field, and the idea extends to dual spaces, where the dual basis corresponds to the original basis via coordinate functionals. Basis vectors thus provide a foundational link between abstract vector spaces and concrete coordinate representations.