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| {{Dablink|See [[scalar (disambiguation)|scalar]] for an account of the broader concept also used in physics and computing.}}
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| In [[linear algebra]], [[real number]]s are called '''scalars''' and relate to vectors in a [[vector space]] through the operation of [[scalar multiplication]], in which a vector can be multiplied by a number to produce another vector.<ref>{{cite book | last=Lay | first=David C. | title=Linear Algebra and Its Applications | publisher=[[Addison–Wesley]] | year=2006 | edition = 3rd | isbn=0-321-28713-4}}</ref><ref>{{cite book | last=Strang | first=Gilbert | authorlink=Gilbert Strang | title=Linear Algebra and Its Applications | publisher=[[Brooks Cole]] | year=2006 | edition = 4th | isbn=0-03-010567-6}}</ref><ref>{{cite book | last = Axler | first = Sheldon | title = Linear Algebra Done Right | publisher = [[Springer Science+Business Media|Springer]] | year = 2002 | edition = 2nd | isbn = 0-387-98258-2}}</ref> More generally, a vector space may be defined by using any [[field (mathematics)|field]] instead of real numbers, such as [[complex number]]s. Then the scalars of that vector space will be the elements of the associated field.
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| A [[inner product|scalar product]] operation (not to be confused with [[scalar multiplication]]) may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an [[inner product space]].
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| The real component of a [[quaternion]] is also called its '''scalar part'''.
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| The term is also sometimes used informally to mean a vector, [[matrix (mathematics)|matrix]], [[tensor]], or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×''n'' matrix and an ''n''×1 matrix, which is formally a 1×1 matrix, is often said to be a '''scalar'''.
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| The term '''[[scalar matrix]]''' is used to denote a matrix of the form ''kI'' where ''k'' is a scalar and ''I'' is the [[identity matrix]].
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| ==Etymology==
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| The word ''scalar'' derives from the Latin word ''scalaris'', adjectival form from ''scala'' ([[Latin language|Latin]] for "ladder"). The English word "scale" is also derived from ''scala''. The first recorded usage of the word "scalar" in mathematics was by [[François Viète]] in ''Analytic Art'' (''In artem analyticen isagoge'')(1591):<ref>http://math.ucdenver.edu/~wcherowi/courses/m4010/s08/lcviete.pdf Lincoln Collins. Biography Paper: Francois Viete</ref>
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| :''Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another are called scalar terms.''
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| :(Latin: ''Magnitudines quae ex genere ad genus sua vi proportionaliter adscendunt vel descendunt, vocentur Scalares.'')
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| According to a citation in the ''[[Oxford English Dictionary]]'' the first recorded usage of the term in English was by [[William Rowan Hamilton|W. R. Hamilton]] in 1846, to refer to the real part of a [[quaternion]]:
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| :''The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part.''
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| ==Definitions and properties==
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| ===Scalars of vector spaces===
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| A [[vector space]] is defined as a set of vectors, a set of scalars, and a [[scalar multiplication]] operation that takes a scalar ''k'' and a vector '''v''' to another vector ''k'''''v'''. For example, in a [[coordinate space]], the scalar multiplication <math>k(v_1, v_2, \dots, v_n)</math> yields <math> (kv_1, kv_2, \dots, k v_n)</math>. In a (linear) [[function space]], ''kƒ'' is the function ''x'' {{Mapsto}} ''k''(''ƒ''(''x'')).
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| The scalars can be taken from any field, including the [[rational number|rational]], [[algebraic number|algebraic]], real, and complex numbers, as well as [[finite field]]s. a number by the elements inside the brackets.
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| ===Scalars as vector components===
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| According to a fundamental theorem of linear algebra, every vector space has a [[basis (linear algebra)|basis]]. It follows that every vector space over a scalar field ''K'' is [[isomorphism|isomorphic]] to a [[coordinate vector space]] where the coordinates are elements of ''K''. For example, every real vector space of [[dimension (vector space)|dimension]] ''n'' is isomorphic to ''n''-dimensional real space '''R'''<sup>''n''</sup>.
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| ===Scalars in normed vector spaces===
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| Alternatively, a vector space ''V'' can be equipped with a [[norm (mathematics)|norm]] function that assigns to every vector '''v''' in ''V'' a scalar ||'''v'''||. By definition, multiplying '''v''' by a scalar ''k'' also multiplies its norm by |''k''|. If ||'''v'''|| is interpreted as the ''length'' of '''v''', this operation can be described as '''scaling''' the length of '''v''' by ''k''. A vector space equipped with a norm is called a [[normed vector space]] (or ''normed linear space'').
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| The norm is usually defined to be an element of ''V''<nowiki>'</nowiki>s scalar field ''K'', which restricts the latter to fields that support the notion of sign. Moreover, if ''V'' has dimension 2 or more, ''K'' must be closed under square root, as well as the four arithmetic operations; thus the rational numbers '''Q''' are excluded, but the [[surd field]] is acceptable. For this reason, not every scalar product space is a normed vector space.
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| ===Scalars in modules===
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| When the requirement that the set of scalars form a field is relaxed so that it need only form a [[ring (mathematics)|ring]] (so that, for example, the division of scalars need not be defined, or the scalars need not be [[commutative]]), the resulting more general algebraic structure is called a [[module (mathematics)|module]].
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| In this case the "scalars" may be complicated objects. For instance, if ''R'' is a ring, the vectors of the product space ''R''<sup>''n''</sup> can be made into a module with the ''n''×''n'' matrices with entries from ''R'' as the scalars. Another example comes from [[manifold|manifold theory]], where the space of [[Section (fiber bundle)|sections]] of the [[tangent bundle]] forms a module over the [[algebra]] of real functions on the manifold.
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| ===Scaling transformation===
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| The scalar multiplication of vector spaces and modules is a special case of [[scaling (geometry)|scaling]], a kind of [[linear transformation]].
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| ===Scalar operations (computer science)===
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| Operations that apply to a single value at a time.
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| * [[Scalar processor]]
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| ==See also==
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| * [[Scalar (physics)]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| * {{springer|title=Scalar|id=p/s083240}}
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| * {{MathWorld |urlname=Scalar |title=Scalar}}
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| * [http://www.mathwords.com/s/scalar.htm Mathwords.com – Scalar]
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| {{Linear algebra}}
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| {{DEFAULTSORT:Scalar (Mathematics)}}
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| [[Category:Abstract algebra]]
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| [[Category:Linear algebra]]
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| [[Category:Analytic geometry]]
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