An n-dimensional vector space (n-space) generalizes the arrow-based vectors of one, two, and three dimensions to an ordered array of n real numbers, called an n-tuple, written x1, x2, ..., xn (abbreviated x-bar). A set of n-tuples becomes an n-dimensional vector space only when three structural definitions are imposed on it by analogy with arrow vectors: equality (component-by-component equality), addition (component-by-component addition), and scalar multiplication (component-by-component scaling); any set of n-tuples obeying these three definitions inherits all properties previously derived for arrows, even though it can no longer be visualized geometrically once n exceeds three. This structure belongs to multivariable calculus and linear algebra, and it underlies the modern treatment of real-valued functions of several real variables, whose domain is an n-tuple and whose codomain is a scalar.
An n-dimensional vector space (n-space) generalizes the arrow-based vectors of one, two, and three dimensions to an ordered array of n real numbers, called an n-tuple, written x1, x2, ..., xn (abbrev…