Editing Basis (linear algebra) (section)

{{short description|Set of vectors used to define coordinates}}
{{redirect|Basis (mathematics)||Basis (disambiguation)#Mathematics{{!}}Basis}}
[[File:3d two bases same vector.svg|130px|thumb|The same vector can be represented in two different bases (purple and red arrows).]]

In [[mathematics]], a [[Set (mathematics)|set]] {{mvar|B}} of elements of  a [[vector space]] {{math|''V''}} is called a '''basis''' ({{plural form}}: '''bases''') if every element of {{math|''V''}} can be written in a unique way as a finite [[linear combination]] of elements of {{mvar|B}}. The coefficients of this linear combination are referred to as '''components''' or '''coordinates''' of the vector with respect to {{mvar|B}}. The elements of a basis are called '''{{visible anchor|basis vectors}}'''.

Equivalently, a set {{mvar|B}} is a basis if its elements are [[linearly independent]] and every element of {{mvar|V}} is a [[linear combination]] of elements of {{mvar|B}}.<ref>{{cite book |last=Halmos |first=Paul Richard |author-link=Paul Halmos |year=1987 |title=Finite-Dimensional Vector Spaces |edition=4th |publisher=Springer |location=New York |url=https://books.google.com/books?id=mdWeEhA17scC&pg=PA10 |page=10 |isbn=978-0-387-90093-3 }}</ref> 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 (vector space)|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.

Basis vectors find applications in the study of [[crystal structure]]s and [[frame of reference|frames of reference]].