In linear algebra, real numbers 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. More generally, a vector
space may be defined by using any field instead of real numbers, such as complex
numbers. Then the scalars of that vector space will be the elements of the
associated field. A 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.
The real component of a quaternion is also called its scalar part.
The term is also sometimes used informally to mean a vector, 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.
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 word scalaris, an adjectival form of
scala. The English word "scale" also comes from scala. The first recorded
usage of the word "scalar" in mathematics occurs in François Viète's
Analytic Art: 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 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 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 kv. For example, in a coordinate space, the scalar multiplication yields
. In a function space, kƒ is the function x ↦ k(ƒ(x)).
The scalars can be taken from any field, including the rational, algebraic, real,
and complex numbers, as well as finite fields. a number by the elements inside
the brackets. = 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 scalar field K is
isomorphic to a coordinate vector space where the coordinates are elements of K.
For example, every real vector space of dimension n is isomorphic to
n-dimensional real space Rn. = 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.
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, the resulting more general algebraic structure is
called a module. In this case the "scalars" may be
complicated objects. For instance, if R is a ring, the vectors of the product
space Rn 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 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, a kind of linear transformation.
= Scalar operations = Operations that apply to a single value
at a time. Scalar processor
See also Scalar
References External links
Hazewinkel, Michiel, ed., "Scalar", Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4 Weisstein, Eric W., "Scalar", MathWorld.
Mathwords.com – Scalar