Scalar (mathematics)

Scalar (mathematics)

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, the scalars associated with a vector space may be complex numbers or elements from any algebraic field.

Also, 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.


The word "scalar" derives from the English word "scale" for a range of numbers, which in turn is derived from "scala" (Latin for "ladder"). According to a citation in the "Oxford English Dictionary" the first recorded usage of the term was by W. R. Hamilton in 1846, to refer 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

calars 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 "k"v. For example, in a coordinate space, the scalar multiplication k(v_1, v_2, ..., v_n) yields (kv_1, kv_2, ..., k v_n). In a (linear) function space, "kf" is the function "x" mapsto "k"("f"("x")).

The scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields.

calars 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.

calar product

A scalar product space is a vector space "V" with an additional scalar product (or "inner product") operation which allows two vectors to be multiplied to produce a number. The result is usually defined to be a member of "V"'s scalar field. Since the inner product of a vector and itself has to be non-negative, a scalar product space can be defined only over fields that support the notion of sign. This excludes finite fields, for instance.

The existence of the scalar product makes it possible to carry geometric intuition over from Euclidean space by providing a well-defined notion of the angle between two vectors, and in particular a way of expressing when two vectors are orthogonal. Most scalar product spaces can also be considered normed vector spaces in a natural way.

calars 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.

calars 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), 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 "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 of the tangent bundle forms a module over the algebra of real functions on the manifold.

caling transformation

The scalar multiplication of vector spaces and modules is a special case of scaling, a kind of linear transformation.

ee also

*Scalar (physics)

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • scalar Mathematics & Physics — [ skeɪlə] adjective (of a quantity) having only magnitude, not direction. noun a scalar quantity. Origin C17: from L. scalaris, from scala (see scale3) …   English new terms dictionary

  • scalar — Mathematics & Physics ► ADJECTIVE ▪ having only magnitude, not direction. ► NOUN ▪ a scalar quantity (especially as opposed to a vector). ORIGIN Latin scalaris, from scala ladder …   English terms dictionary

  • Scalar (physics) — In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations (in Newtonian mechanics), or by Lorentz transformations or space time translations (in relativity). (Contrast to… …   Wikipedia

  • Scalar — A scalar is a variable that only has magnitude, e.g. a speed of 40 km/h. Compare it with vector, a quantity comprising both magnitude and direction, e.g. a velocity of 40km/h north .* Scalar (mathematics), a quantity which is independent of… …   Wikipedia

  • Scalar field — In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature… …   Wikipedia

  • Scalar curvature — In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the… …   Wikipedia

  • 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). Note that scalar multiplication is different from scalar product which is an inner… …   Wikipedia

  • Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia

  • Scalar field theory — In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a scalar , in contrast to a vector or tensor field. The quanta of the… …   Wikipedia

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”