Math 2.0

Linear and multilinear algebra; matrix theory Definition

Dimension

Builds on BasisFinite-dimensional vector space

Definition

Let FF be a field and let VV be a finite-dimensional vector space over FF. The dimension of VV over FF, written dim⁡FV\dim_F V or dim⁡V\dim V, is the number of vectors in a basis of VV. For an arbitrary vector space, the dimension is the cardinality of a basis.

Remarks

The number is well defined, by the dimension theorem for vector spaces. The zero space {0}\{0\} has the empty basis and dimension 00. The dimension depends on the field: dim⁡ℂℂ=1\dim_\C \C = 1, but dim⁡ℝℂ=2\dim_\R \C = 2. Mathlib writes Module.finrank F V for the dimension as a natural number, with value 00 for infinite-dimensional spaces.

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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 levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinition
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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. Finite-dimensional vector space definition
  9. Dimension definition

Where this leads

26 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), lead section
About this page
Tag05G2G0P224
URIm2://05G2G0P224
Kinddefinition
WritingW0 stub
MSC15A03
DefinesDimension
UsesBasis, Finite-dimensional vector space

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