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 scalar 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.