Math 2.0

statement theorem

Existence of an orthonormal basis

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.

Explanations

L1 human

Every finite-dimensional inner product space has a basis of perpendicular unit vectors. We construct one with the Gram–Schmidt process. So every such space looks like Rn\R^n or Cn\C^n with the standard inner product.

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To read this node, read these 13 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. Gram–Schmidt process concept
  12. Orthonormal basis definition
  13. Existence of an orthonormal basis statement

Sources

  1. Web https://en.wikipedia.org/wiki/Orthonormal_basis, Existence

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