View source for Scalar (mathematics)

{{short description|Elements of  a field, e.g. real numbers, in the context of linear algebra}}
{{distinguish|Scalar (physics)}}
{{Other uses|Scalar (disambiguation){{!}}Scalar}}
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A '''scalar''' is an element of a [[Field (mathematics)|field]] which is used to define a ''[[vector space]]''.
In [[linear algebra]], [[real number]]s or generally elements of a field are called '''scalars''' and relate to vectors in an associated vector space through the operation of [[scalar multiplication]] (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.<ref>{{cite book |last=Lay |first=David C. |title=Linear Algebra and Its Applications |url=https://archive.org/details/studyguidetoline0000layd |url-access=registration |publisher=[[Addison–Wesley]] |year=2006 |edition=3rd |isbn=0-321-28713-4}}</ref><ref>{{cite book |last=Strang |first=Gilbert |author-link=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> Generally speaking, a vector space may be defined by using any field instead of real numbers (such as [[complex number]]s). Then scalars of that vector space will be elements of the associated field (such as complex numbers).

A [[Inner product|scalar product]] operation –&nbsp;not to be confused with scalar multiplication&nbsp;– may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an [[inner product space]].

A quantity described by multiple scalars, such as having both direction and magnitude, is called a ''[[vector (mathematics and physics)|vector]]''.<ref>[http://www.mathwords.com/s/scalar.htm Mathwords.com – Scalar]</ref>
The term scalar 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&thinsp;×&thinsp;''n'' matrix and an ''n''&thinsp;×&thinsp;1 matrix, which is formally a 1&thinsp;×&thinsp;1 matrix, is often said to be a '''scalar'''.
The real component of a [[quaternion]] is also called its '''scalar part'''.

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]].

==Etymology==

The word ''scalar'' derives from the [[Latin language|Latin]] word ''scalaris'', an adjectival form of ''scala'' (Latin for "ladder"), from which the English word ''scale'' also comes. The first recorded usage of the word "scalar" in mathematics occurs in [[François Viète]]'s ''Analytic Art'' (''In artem analyticem isagoge'') (1591):<ref>
{{cite book
| last1                 = Vieta
| first1                = Franciscus
| author-link1          = François Viète
| title                 = In artem analyticem isagoge seorsim excussa ab Opere restitutae mathematicae analyseos, seu Algebra noua
|trans-title=Guide to the analytic art [...] or new algebra
| url                   = https://books.google.com/books?id=BWTyywN39KEC
| language              = la
| location              = Tours
| page                  = 4 ([[verso]])
| publisher             = apud Iametium Mettayer typographum regium
| date      = 1591
| access-date            = 2015-06-24
}}
</ref><ref>
http://math.ucdenver.edu/~wcherowi/courses/m4010/s08/lcviete.pdf Lincoln Collins. Biography Paper: Francois Viete</ref>

:Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another may be called scalar terms.
:(Latin: ''Magnitudines quae ex genere ad genus sua vi proportionaliter adscendunt vel descendunt, vocentur Scalares.'')

According to a citation in the ''[[Oxford English Dictionary]]'' the first recorded usage of the term "scalar" in English came with [[William Rowan Hamilton|W. R. Hamilton]] in 1846, referring to the real part of a quaternion:

:''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.''

==Definitions and properties==
[[File:Vector components.svg|thumb|Scalars are [[real numbers]] used in linear algebra, as opposed to [[vector (mathematics and physics)|vectors]]. This image shows a [[Euclidean vector]]. Its coordinates ''x'' and ''y'' are scalars, as is its length, but '''v''' is not a scalar.]]
===Scalars of vector spaces===
A [[vector space]] is defined as a set of vectors (additive [[abelian group]]), a set of scalars ([[Field (mathematics)|field]]), and a scalar multiplication operation that takes a scalar ''k'' and a vector '''v''' to form 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> (k v_1, k v_2, \dots, k v_n)</math>. In a (linear) [[function space]], {{math|''kf''}} is the function {{math|''x'' {{Mapsto}} ''k''(''f''(''x''))}}.

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.

===Scalars as vector components===
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 field ''K'' is [[isomorphism|isomorphic]] to the corresponding [[coordinate vector space]] where each [[coordinate]] consists of elements of ''K'' (E.g., coordinates (''a<sub>1</sub>'', ''a<sub>2</sub>'', ..., ''a<sub>n</sub>'') where ''a<sub>i</sub>'' ∈ ''K'' and ''n'' is the dimension of the vector space in consideration.). For example, every real vector space of [[dimension (vector space)|dimension]] ''n'' is isomorphic to the ''n''-dimensional real space '''R'''<sup>''n''</sup>.

===Scalars in normed vector spaces===
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'').

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 [[quadratic surd|surd field]] is acceptable. For this reason, not every scalar product space is a normed vector space.

===Scalars in modules===
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]].

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''&nbsp;×&nbsp;''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.

===Scaling transformation===
The scalar multiplication of vector spaces and modules is a special case of [[scaling (geometry)|scaling]], a kind of [[linear transformation]].

==See also==
* [[Algebraic structure]]
* [[Scalar (physics)]]
*[[Linear algebra]]
*[[Matrix (mathematics)]]
*[[Row and column vectors]]
*[[Tensor]]
*[[Vector (mathematics and physics)]]
*[[Vector calculus]]

==References==
{{reflist}}

==External links==
{{Commonscat|Scalar (mathematics)}}
*{{springer|title=Scalar|id=p/s083240}}
*{{MathWorld |urlname=Scalar |title=Scalar}}
*[http://www.mathwords.com/s/scalar.htm Mathwords.com – Scalar]

{{Linear algebra}}

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[[Category:Scalars|*]]
[[Category:Linear algebra]]