Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of a subspace

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. Then WW is finite-dimensional, and dim⁡W≤dim⁡V\dim W \le \dim V, where dim⁡\dim is the dimension.

Remarks

A basis of WW is a linearly independent family in VV, so it has at most dim⁡V\dim V vectors. Equality holds only when W=VW = V, see a subspace of full dimension is the whole space. For infinite-dimensional VV, the inequality dim⁡W≤dim⁡V\dim W \le \dim V of cardinals still holds.

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 levelDimensiondefinitionDimension of a subspace, statement, no evidence levelDimension of a subspa…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. Dimension of a subspace statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties
About this page
Tag0QW084460E
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Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0QW084460E.
UsesDimension, Linear subspace

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