Vector space

A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.

Subspaces and bases

Main articles: Linear subspace, Basis

Related Topics:
Linear subspace - Basis

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Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is described as being linearly independent. A linearly independent set whose span is the whole space is called a basis.

Related Topics:
Subspace - Span - Linearly independent - Basis

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All bases for a given vector space have the same cardinality. Using Zorn’s Lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance, the real vector spaces are just R0, R1, R2, R3, …, R∞, …. As you would expect, the dimension of the real vector space R3 is three.

Related Topics:
Cardinality - Zorn’s Lemma - Isomorphism - Cardinal number - Dimension

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A basis makes it possible to express every vector of the space as a unique combination of the field elements. Vector spaces are usually introduced from this coordinatised viewpoint.

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Given a translationally invariant and rescaling invariant topology over a vector space (preferably infinite-dimensional), the sum of an infinite sequence of vectors can be defined as the topological limit, if it exists. See topological vector space.

Related Topics:
Topology - Infinite sequence - Topological limit - Topological vector space

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