Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A subspace of full dimension is the whole space

Builds on DimensionLinear subspace

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a linear subspace of VV. If dim⁡W=dim⁡V\dim W = \dim V, then W=VW = V.

Remarks

A basis of WW is a linearly independent family of dim⁡V\dim V vectors in VV. So it is a basis of VV, see linearly independent set of the right size is a basis. The statement fails for infinite-dimensional spaces. In F[x]F[x], the subspace of polynomials with p(0)=0p(0) = 0 has the same dimension as F[x]F[x], but it is not equal to F[x]F[x].

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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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 levelDimensiondefinitionA subspace of full dimension is the whole space, statement, no evidence levelA subspace of full di…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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
  10. A subspace of full dimension is the whole space statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties
About this page
Tag0WGAXEK99E
URIm2://0WGAXEK99E
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0WGAXEK99E.
UsesDimension, Linear subspace

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