Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Coordinates in an orthonormal basis

Builds on Orthonormal basis

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, and let (e1,…,en)(e_1, \dots, e_n) be an orthonormal basis of VV. Then, for each v∈Vv \in V, v=∑i=1n⟨v,ei⟩ ei.v = \sum_{i=1}^{n} \langle v, e_i \rangle\, e_i. So the coordinates of vv in this basis are the inner products ⟨v,ei⟩\langle v, e_i \rangle.

Remarks

If v=∑jajejv = \sum_j a_j e_j, then ⟨v,ei⟩=∑jaj⟨ej,ei⟩=ai\langle v, e_i \rangle = \sum_j a_j \langle e_j, e_i \rangle = a_i. For the basis 12(1,1)\frac{1}{\sqrt 2}(1, 1), 12(1,−1)\frac{1}{\sqrt 2}(1, -1) of ℝ2\R^2, the vector (3,1)(3, 1) has the coordinates 4/24/\sqrt 2 and 2/22/\sqrt 2. With the convention of linearity in the second argument, the coefficient is written ⟨ei,v⟩\langle e_i, v \rangle.

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 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 basisdefinitionCoordinates in an orthonormal basis, statement, no evidence levelCoordinates in an ort…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. Orthonormal family definition
  9. Spanning set definition
  10. Basis definition
  11. Orthonormal basis definition
  12. Coordinates in an orthonormal basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthonormal_basis, Basic formula
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Kindstatement, proposition
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MSC15A63, 46C05
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UsesOrthonormal basis

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