A scalar is an element of a
field 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 matrice ...
,
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 (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 fo ...
s). Then scalars of that vector space will be elements of the associated field (such as complex numbers).
A
scalar product 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''.
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'' (franchi ...
,
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 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 languages, 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 ...
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):
[
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 c ...
'' 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), 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
yields
. In a (linear)
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
, is the function .
The scalars can be taken from any field, including the
rational,
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
, 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. It follows that every vector space over a field ''K'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the corresponding
coordinate vector space where each
coordinate consists of elements of ''K'' (E.g., coordinates (''a
1'', ''a
2'', ..., ''a
n'') where ''a
i'' ∈ ''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 coord ...
''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 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''
'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 (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
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
* Modul ...
.
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, 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 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 ...
of real functions on the manifold.
Scaling transformation
The scalar multiplication of vector spaces and modules is a special case of
scaling, a kind of
linear transformation.
See also
*
Algebraic structure
*
Scalar (physics)
*
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 matrice ...
References
External links
*
*
Mathwords.com – Scalar
{{DEFAULTSORT:Scalar (Mathematics)
*
Linear algebra