Math 2.0

statement theorem

Dimension theorem for vector spaces

Statement

Let FF be a field and let VV be a vector space over FF. Any two bases of VV have the same cardinality.

Remarks

For finite bases, this follows from the Steinitz exchange lemma. For infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis. The common cardinality is the dimension of VV. The theorem holds more generally for free modules over a commutative ring that is not the zero ring.

Explanations

L1 human

However we choose a basis of a space, it always has the same number of vectors. So "the number of vectors in a basis" is a property of the space itself. We call this number the dimension.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionDimension theorem for vector spaces, statement, no evidence levelDimension theorem for…statement
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  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Spanning set definition
  7. Basis definition
  8. Dimension theorem for vector spaces statement

Sources

  1. Web https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces, lead section

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