Talk:Basis (linear algebra)

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Revision as of 14:44, 25 November 2008 by imported>Richard Pinch (→‎Some points: new section)
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 Definition:  A set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others. [d] [e]
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 Workgroup category:  Mathematics [Categories OK]
 Talk Archive:  none  English language variant:  American English

Some points

  • "every vector in V can be written uniquely as a finite linear combination of vectors in the basis". Is it necessary to say finite here?
  • "Every nonzero vector space has a basis". Why non-zero? The zero space has the empty set as basis.
  • "Every nonzero vector space has a basis". Strictly speaking this requires Axiom of Choice.
  • "and in fact, infinitely many different bases". The first clue that these are real vector spaces.
  • "every finite dimensional vector space can be considered to be essentially 'the same as' a Euclidean space". I disagree, Euclidean space has a metric, which general vector spaces don't, and the different coordinate mappings will induce different metrics back on the vector space.
  • It needs to be stated that the number of elements in a basis is always the same, hence the definition of dimension.

Richard Pinch 19:43, 25 November 2008 (UTC)

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