Basis (linear algebra)

In linear algebra, a basis for a vector space $$V$$ is a set of vectors in $$V$$ such that every vector in $$V$$ can be written uniquely as a finite linear combination of vectors in the space. Every nonzero vector space has a basis, and this fact is the foundation for much of the structure of the theory of vector spaces. For instance, a basis for a finite-dimensional vector space provides the space with a invertible linear transformation to Euclidean space, given by taking the coordinates of a vector with respect to a basis. Through this transformation, every finite dimensional vector space can be considered to be essentially "the same as" a Euclidean space, just with different labels for the vectors and operations.