Math 2.0

Linear and multilinear algebra; matrix theory Definition

Orthonormal basis

Builds on Orthonormal familyBasis

Definition

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C. An orthonormal basis of VV is a basis (e1,…,en)(e_1, \dots, e_n) of VV that is an orthonormal family.

Remarks

An orthonormal family of dim⁡V\dim V vectors is an orthonormal basis. In an orthonormal basis, coordinates are inner products: v=∑i⟨v,ei⟩eiv = \sum_i \langle v, e_i \rangle e_i. The vectors 12(1,1)\frac{1}{\sqrt 2}(1, 1) and 12(1,−1)\frac{1}{\sqrt 2}(1, -1) form an orthonormal basis of ℝ2\R^2. In infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.

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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 spandefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinition
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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. Orthonormal family definition
  9. Spanning set definition
  10. Basis definition
  11. Orthonormal basis definition

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthonormal_basis, lead section
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MSC15A63, 46C05
DefinesOrthonormal basis
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