Talk:Basis (linear algebra): Difference between revisions
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imported>Barry R. Smith (New page: {{subpages}}) |
imported>Richard Pinch (→Some points: new section) |
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== 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. | |||
[[User:Richard Pinch|Richard Pinch]] 19:43, 25 November 2008 (UTC) | |||
Revision as of 13:44, 25 November 2008
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)