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Basis (linear algebra)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> B of elements of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> <i>V</i> is called a basis (pl.: bases) if every element of <i>V</i> can be written in a unique way as a finite <a href="/facts/Linear_combination/CVjL6kuN">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 basis vectors. </p><p>Equivalently, a set B is a basis if its elements are <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> and every element of V is a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of elements of B. In other words, a basis is a linearly independent <a href="/facts/Spanning_set/vPjclEtb">spanning set</a>. </p><p>A vector space can have several bases; however all the bases have the same number of elements, called the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of the vector space. </p><p>This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. </p><p>Basis vectors find applications in the study of <a href="/facts/Crystal_structure/bJ4bCeHd">crystal structures</a> and <a href="/facts/Frame_of_reference/15yYfB9M">frames of reference</a>. </p>