Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Gram–Schmidt vectors are orthogonal

Builds on Gram–Schmidt processOrthogonal vectors

Statement

Let VV be an inner product space over ℝ\R or ℂ\C, and let v1,…,vk∈Vv_1, \dots, v_k \in V be linearly independent. Let u1,…,uku_1, \dots, u_k be the vectors of the Gram–Schmidt process. Then u1,…,uku_1, \dots, u_k are nonzero and pairwise orthogonal, and span⁡(u1,…,uj)=span⁡(v1,…,vj)(1≤j≤k).\operatorname{span}(u_1, \dots, u_j) = \operatorname{span}(v_1, \dots, v_j) \qquad (1 \le j \le k).

Remarks

The proof is by induction on jj: by construction, ⟨uj,ui⟩=0\langle u_j, u_i \rangle = 0 for i<ji < j. The vector uju_j is nonzero, because vjv_j is not in span⁡(v1,…,vj−1)\operatorname{span}(v_1, \dots, v_{j-1}). If the viv_i are only spanning, then the process produces some zero vectors, and dropping them still gives an orthogonal family with the same span.

Reading path

To understand this page from the ground up, read these 10 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 familydefinitionGram–Schmidt process, concept, no evidence levelGram–Schmidt processconceptGram–Schmidt vectors are orthogonal, statement, no evidence levelGram–Schmidt vectors …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. Gram–Schmidt process concept
  10. Gram–Schmidt vectors are orthogonal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gram–Schmidt_process, Properties
About this page
Tag0WXBGBTA3D
URIm2://0WXBGBTA3D
Kindstatement, proposition
WritingW0 stub
MSC15A63, 65F25
StatusThis copy of the site has no build data. Run m2 status 0WXBGBTA3D.
UsesGram–Schmidt process, Orthogonal vectors

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