Vector space

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

Linear transformations

Main article: Linear transformation

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Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted L(V, W), is also a vector space over F. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices.

Related Topics:
Linear transformation - Matrices

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An isomorphism is a linear map that is one-to-one and onto. If there exists an isomorphism between V and W, we call the two spaces isomorphic; they are then essentially identical.

Related Topics:
Isomorphism - One-to-one - Onto

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The vector spaces over a fixed field F, together with the linear maps, form a category.

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