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Vector Space

Basic definition

A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space.

Detailed definition

A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R^n, where every element is represented by a list of n real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately. For a general vector space, the scalars are members of a field F, in which case V is called a vector space over F. Euclidean n-space R^n is called a real vector space, and C^n is called a complex vector space.

Related terms

Banach space | field | function space | Hilbert space | inner product space | module | quotient vector space | ring | symplectic space | topological vector space | vector | vector basis

Educational grade level

college level

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