Math 2.0

Linear and multilinear algebra; matrix theory Vector spaces Proposition

Dimension of a quotient space

Builds on Quotient spaceDimension

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a subspace of VV. Then the quotient space V/WV / W is finite-dimensional, and dim⁡(V/W)+dim⁡W=dim⁡V,\dim(V / W) + \dim W = \dim V, where dim⁡\dim is the dimension.

Remarks

The number dim⁡(V/W)=dim⁡V−dim⁡W\dim(V / W) = \dim V - \dim W is the codimension of WW in VV. The formula is the rank–nullity theorem for the quotient map V→V/WV \to V / W. For example, a plane WW in ℝ3\R^3 gives dim⁡(ℝ3/W)=1\dim(\R^3 / W) = 1.

Reading path

To understand this page from the ground up, read these 11 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 spandefinitionQuotient space, definition, no evidence levelQuotient spacedefinitionSpanning 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 quotient space, statement, no evidence levelDimension of a quotie…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. Quotient space definition
  7. Spanning set definition
  8. Basis definition
  9. Finite-dimensional vector space definition
  10. Dimension definition
  11. Dimension of a quotient space statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra), Properties
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Kindstatement, proposition
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MSC15A03
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UsesQuotient space, Dimension

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