scalar (mathematics)
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A scalar is an element of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
which is used to define a ''
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
(defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
operation – not to be confused with scalar multiplication – 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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. A quantity described by multiple scalars, such as having both direction and magnitude, is called a ''
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
''. The term scalar is also sometimes used informally to mean a vector,
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
,
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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. The real component of a quaternion is also called its scalar part. The term
scalar matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
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 Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
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 François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
's ''Analytic Art'' (''In artem analyticem isagoge'') (1591): http://math.ucdenver.edu/~wcherowi/courses/m4010/s08/lcviete.pdf Lincoln Collins. Biography Paper: Francois Viete :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 ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a com ...
'' the first recorded usage of the term "scalar" in English came with 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


Scalars of vector spaces

A
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
is defined as a set of vectors (additive
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
), a set of scalars (
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
), 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, the scalar multiplication k(v_1, v_2, \dots, v_n) yields (k v_1, k v_2, \dots, k v_n). In a (linear) function space, is the function . The scalars can be taken from any field, including the
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
, algebraic, real, and complex numbers, as well as
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s.


Scalars as vector components

According to a fundamental theorem of linear algebra, every vector space has a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
. It follows that every vector space over a field ''K'' is isomorphic to the corresponding
coordinate vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
where each
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
consists of elements of ''K'' (E.g., coordinates (''a1'', ''a2'', ..., ''an'') where ''ai'' ∈ ''K'' and ''n'' is the dimension of the vector space in consideration.). For example, every real vector space of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
''n'' is isomorphic to the ''n''-dimensional real space R''n''.


Scalars in normed vector spaces

Alternatively, a vector space ''V'' can be equipped with a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
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 In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(or ''normed linear space''). The norm is usually defined to be an element of ''V'''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.


Scalars in modules

When the requirement that the set of scalars form a field is relaxed so that it need only form a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
(so that, for example, the division of scalars need not be defined, or the scalars need not be
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
), the resulting more general algebraic structure is called a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
. In this case the "scalars" may be complicated objects. For instance, if ''R'' is a ring, the vectors of the product space ''R''''n'' can be made into a module with the ''n''×''n'' matrices with entries from ''R'' as the scalars. Another example comes from
manifold theory In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
, where the space of
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
forms a module over the
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
of real functions on the manifold.


Scaling transformation

The scalar multiplication of vector spaces and modules is a special case of
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
, a kind of
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
.


See also

*
Algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
*
Scalar (physics) In physics, scalars (or scalar quantities) are physical quantities that are unaffected by changes to a vector space basis (i.e., a coordinate system transformation). Scalars are often accompanied by units of measurement, as in "10 cm". Examples o ...
*
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...


References


External links

* *
Mathwords.com – Scalar
{{DEFAULTSORT:Scalar (Mathematics) * Linear algebra