Set of vectors used to define coordinates

Basis (linear algebra)

Basis (linear algebra)

In <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a>, a <a href="/en/Set_(mathematics)" title="Set (mathematics)" class="wl">set</a> B of vectors in a <a href="/en/Vector_space" title="Vector space" class="wl">vector space</a> V is called a basis if every element of V may be written in a unique way as a finite <a href="/en/Linear_combination" title="Linear combination" class="wl">linear combination</a> 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 <style data-mw-deduplicate="TemplateStyles:r1023754711" typeof="mw:Extension/templatestyles mw:Transclusion">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}</style>basis vectors.

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Set of vectors used to define coordinates</div>
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<figure class="wiki-thumb"><a href="/en/File:3d_two_bases_same_vector.svg" class="mw-file-description image media-thumb" data-hash="File:3d_two_bases_same_vector.svg"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/3d_two_bases_same_vector.svg/130px-3d_two_bases_same_vector.svg.png" decoding="async" data-file-width="161" data-file-height="284" data-file-type="drawing" height="229" width="130" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/3d_two_bases_same_vector.svg/195px-3d_two_bases_same_vector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/3d_two_bases_same_vector.svg/260px-3d_two_bases_same_vector.svg.png 2x" alt="3d_two_bases_same_vector.svg"></a><figcaption>The same vector can be represented in two different bases (purple and red arrows).</figcaption></figure>
Equivalently, a set B is a basis if its elements are <a href="/en/Linearly_independent" title="Linearly independent" class="mw-redirect wl">linearly independent</a> and every element of V is a <a href="/en/Linear_combination" title="Linear combination" class="wl">linear combination</a> of elements of B.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a> In other words, a basis is a linearly independent <a href="/en/Spanning_set" title="Spanning set" class="mw-redirect wl">spanning set</a>.

A vector space can have several bases; however all the bases have the same number of elements, called the <a href="/en/Dimension_(vector_space)" title="Dimension (vector space)" class="wl">dimension of the vector space</a>.

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

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 .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}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. 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. 