# Basis (linear algebra)

In mathematics, a set B of vectors in a vector space *V* is called a **basis** if every element of *V* may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as **components** or **coordinates** of the vector with respect to B. The elements of a basis are called **basis vectors**.

Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.[1] In other words, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the *dimension* of the vector space.

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.