Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

Then scalars of that vector space will be elements of the associated field (such as complex numbers).

A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

[4] The term scalar is also sometimes used informally to mean a vector, matrix, tensor, or other, usually, "compound" value that is actually reduced to a single component.

The real component of a quaternion is also called its scalar part.

The first recorded usage of the word "scalar" in mathematics occurs in François Viète's Analytic Art (In artem analyticem isagoge) (1591):[5][6] 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: A vector space is defined as a set of vectors (additive abelian group), a set of scalars (field), and a scalar multiplication operation that takes a scalar k and a vector v to form another vector kv.

According to a fundamental theorem of linear algebra, every vector space has a basis.

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 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.

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