Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Existence of an orthonormal basis

Builds on Orthonormal basisGram–Schmidt process

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C. Then VV has an orthonormal basis. Moreover, each orthonormal family in VV extends to an orthonormal basis of VV.

Remarks

The proof applies the Gram–Schmidt process to any basis of VV. For the extension, first extend the orthonormal family to a basis, and then apply Gram–Schmidt; it keeps the first vectors. Hence each finite-dimensional real inner product space is isometric to ℝn\R^n with the dot product.

Reading path

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

Show the path as a graph
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 spandefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionGram–Schmidt process, concept, no evidence levelGram–Schmidt processconceptOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionExistence of an orthonormal basis, statement, no evidence levelExistence of an ortho…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. Inner product definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Orthogonal vectors definition
  6. Linear independence definition
  7. Linear span definition
  8. Orthonormal family definition
  9. Spanning set definition
  10. Basis definition
  11. Gram–Schmidt process concept
  12. Orthonormal basis definition
  13. Existence of an orthonormal basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthonormal_basis, Existence
About this page
Tag0Y85GYEGSF
URIm2://0Y85GYEGSF
Kindstatement, theorem
WritingW0 stub
MSC15A63, 46C05
StatusThis copy of the site has no build data. Run m2 status 0Y85GYEGSF.
UsesOrthonormal basis, Gram–Schmidt process

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