Math 2.0

Linear and multilinear algebra; matrix theory Vector spaces Theorem

Grassmann's formula

Builds on Sum of subspacesIntersection of subspacesDimension

Statement

Let FF be a field, let VV be a vector space over FF, and let UU and WW be finite-dimensional linear subspaces of VV. Then dim⁡(U+W)+dim⁡(U∩W)=dim⁡U+dim⁡W,\dim(U + W) + \dim(U \cap W) = \dim U + \dim W, where U+WU + W is the sum of the subspaces and dim⁡\dim is the dimension.

Remarks

The proof extends a basis of U∩WU \cap W to a basis of UU and to a basis of WW. Together, these vectors form a basis of U+WU + W. Another proof applies the rank–nullity theorem to the map U×W→VU \times W \to V, (u,w)↦u−w(u, w) \mapsto u - w. Two planes through the origin in ℝ3\R^3 with U≠WU \ne W have dim⁡(U+W)=3\dim(U + W) = 3, so dim⁡(U∩W)=1\dim(U \cap W) = 1.

Reading path

To understand this page from the ground up, read these 12 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 spandefinitionIntersection of subspaces, statement, no evidence levelIntersection of subsp…statementSum of subspaces, definition, no evidence levelSum of subspacesdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionGrassmann's formula, statement, no evidence levelGrassmann's formulastatement
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. Intersection of subspaces statement
  7. Sum of subspaces definition
  8. Spanning set definition
  9. Basis definition
  10. Finite-dimensional vector space definition
  11. Dimension definition
  12. Grassmann's formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties
About this page
Tag06HTJNCMPA
URIm2://06HTJNCMPA
Aliasesdimension-formula
Kindstatement, theorem
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 06HTJNCMPA.
UsesSum of subspaces, Intersection of subspaces, Dimension

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