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. Linear algebra is the branch of Mathematics concerned with In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in

More generally, the scalars associated with a vector space may be complex numbers or elements from any algebraic field. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

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. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. A vector space equipped with a scalar product is called a inner product space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

The real component of a quaternion is also called its scalar part. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

Etymology

The word scalar derives from the English word "scale" for a range of numbers, which in turn is derived from scala (Latin for "ladder"). Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. 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. The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in For example, in a coordinate space, the scalar multiplication k(v₁,v₂,. In Mathematics, specifically in Linear algebra, the coordinate space, F n, is the prototypical example of an n -dimensional . . ,v_n) yields (kv₁,kv₂,. . . ,kv_n). In a (linear) function space, kf is the function x k(f(x)). In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

The scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

Scalars as vector components

According to a fundamental theorem of linear algebra, every vector space has a basis. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, specifically in Linear algebra, the coordinate space, F n, is the prototypical example of an n -dimensional For example, every real vector space of dimension n is isomorphic to n-dimensional real space Rⁿ. In Mathematics, the dimension of a Vector space V is the cardinality (i

Scalar 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. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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. A negative number is a Number that is less than zero, such as −2 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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Mathematics, two Vectors are orthogonal if they are Perpendicular, i Most scalar product spaces can also be considered normed vector spaces in a natural way. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to

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||. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length 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). In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to

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), the resulting more general algebraic structure is called a module. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.

Scaling transformation

The scalar multiplication of vector spaces and modules is a special case of scaling, a kind of linear transformation. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

See also

• Scalar (physics)