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Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Vector spaces

Dimension

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If vector space $ V$ has a basis consisting of a finite number of vectors $ \{b_1, \ldots ,b_n\}$, then $ n$ is called the dimension of $ V$ (or $ V$ is said to have dimension $ n$)(notation: $ {\rm dim} V = n$).

If $ V = 0$, that is, the only element in $ V$ is the zero vector, then we set $ {\rm dim} V = 0 .$

If a vector space has no finite basis, then it is called infinite-dimensional (notation: $ {\rm dim} V = \infty$).

Observe that according to the general Basis Teorem every vector space has a basis.

All bases of a finite-dimensional vector space have the same length, that is, the same number of basis vectors.

There exist bijections between different bases of a given infinite-dimensional vector space.

(Authors: App/Burkhardt/Höllig/Kimmerle)

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  automatically generated 1/9/2017