TheInfoList

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 grassl ...
which is used to define a ''
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''. A quantity described by multiple scalars, such as having both direction and magnitude, is called a ''
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
''. 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 matrix (mat ...
,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

(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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The real component of a
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

is also called its scalar part. The term scalar is also sometimes used informally to mean a vector,
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
,
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, 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 term
scalar matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonalIn linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix A is the collec ...
is used to denote a matrix of the form ''kI'' where ''k'' is a scalar and ''I'' is the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

.

# 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 spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...
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) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
'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 principal historical dictionary A historical dictionary or dictionary on historical principles is a dictionary which deals not only with the latterday meanings of words but also the historica ...
'' the first recorded usage of the term "scalar" in English came with
W. R. Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, Trinity College Dubli ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is defined as a set of vectors (additive
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
), 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 grassl ...
), 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 $k\left(v_1, v_2, \dots, v_n\right)$ yields $\left(kv_1, kv_2, \dots, k v_n\right)$. In a (linear)
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, ''kƒ'' is the function ''x'' ''k''(''ƒ''(''x'')). The scalars can be taken from any field, including the
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
, algebraic, real, and complex numbers, as well as
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
. It follows that every vector space over a field ''K'' is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to the corresponding coordinate vector space where each coordinates 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
''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 Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
), 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 * Modula ...
. 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 , where the space of sections of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

forms a module over the
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

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, energy ...
, a kind of
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

## Scalar operations (computer science)

Operations that apply to a single value at a time. *
Scalar processor Scalar processors represent a class of computer processor A central processing unit (CPU), also called a central processor, main processor or just Processor (computing), processor, is the electronic circuitry that executes Instruction (computi ...
vs.
vector processor In computing, a vector processor or array processor is a central processing unit (CPU) that implements an instruction set where its Instruction (computer science), instructions are designed to operate efficiently and effectively on large Array da ...
or
superscalar processor A superscalar processor is a CPU A central processing unit (CPU), also called a central processor, main processor or just processor, is the electronic circuit File:PExdcr01CJC.jpg, 200px, A circuit built on a printed circuit board (PCB) ...
*
Variable (computer science) In computer programming Computer programming is the process of designing and building an executable In computing, executable code, an executable file, or an executable program, sometimes simply referred to as an executable or binary, cause ...
sometimes also referred to as a "scalar"

*
Algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Scalar (physics) In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
*
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 matrix (mat ...