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In Linear Algebra , Real Number s 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 Number s 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 ''k''v. For example, in a Coordinate Space , the scalar multiplication yields . In a (linear) Function Space , ''kf'' is the function ''x'' ''k''(''f''(''x'')). The scalars can be taken from any field, including the Rational , Algebraic , real, and complex numbers, as well as Finite Field s. 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 Space s in a natural way. Scalars in normed vector spaces |
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