Math 2.0

Linear and multilinear algebra; matrix theory Corollary

Dimension of the orthogonal complement

Builds on Orthogonal complementDimension

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, and let WW be a subspace of VV. Then dim⁡W+dim⁡W⊥=dim⁡V,\dim W + \dim W^\perp = \dim V, where W⊥W^\perp is the orthogonal complement of WW.

Remarks

It follows from V=W⊕W⊥V = W \oplus W^\perp. For example, in ℝ3\R^3, a line has a plane as orthogonal complement: 1+2=31 + 2 = 3. For a real matrix, it gives rank⁡A+dim⁡ker⁡A=n\rank A + \dim \ker A = n once more, because ker⁡A\ker A is the orthogonal complement of the row space.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionSpanning 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 the orthogonal complement, statement, no evidence levelDimension of the orth…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.
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  1. Vector space definition
  2. Inner product definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Orthogonal vectors definition
  6. Linear independence definition
  7. Linear span definition
  8. Orthogonal complement definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Dimension of the orthogonal complement statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthogonal_complement, Finite dimensions
About this page
Tag0E2NYPVSP9
URIm2://0E2NYPVSP9
Kindstatement, corollary
WritingW0 stub
MSC15A03, 46C05
StatusThis copy of the site has no build data. Run m2 status 0E2NYPVSP9.
UsesOrthogonal complement, Dimension

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