Base field

A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

In linear algebra, real numbers or other elements of a field 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.

The word scalar derives from the Latin word scalaris, an adjectival form of scala (Latin for "ladder"). The English word also comes from scala. The first recorded usage of the word "scalar" in mathematics occurs in François Viète's Analytic Art (In artem analyticem isagoge) (1591):

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