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 a 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 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

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 ${\displaystyle k(v_{1},v_{2},...,v_{n})}$ yields ${\displaystyle (kv_{1},kv_{2},...,kv_{n})}$. In a (linear) function space, kf is the function x ${\displaystyle \mapsto }$ k(f(x)).

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

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.

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. 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.

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), 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.