Math 2.0

Linear and multilinear algebra; matrix theory Proposition

An orthonormal family is linearly independent

Builds on Orthonormal familyLinear independence

Statement

Let VV be an inner product space over ℝ\R or ℂ\C. Then each orthonormal family in VV is linearly independent.

Remarks

If ∑iaiei=0\sum_i a_i e_i = 0, then the inner product with eje_j gives aj=0a_j = 0. More generally, a family of nonzero pairwise orthogonal vectors is linearly independent. So an orthonormal family in an nn-dimensional space has at most nn vectors.

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To understand this page from the ground up, read these 7 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 combinationdefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionAn orthonormal family is linearly independent, statement, no evidence levelAn orthonormal family…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. Orthogonal vectors definition
  5. Linear independence definition
  6. Orthonormal family definition
  7. An orthonormal family is linearly independent statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthonormality, Properties
About this page
Tag0JWM6GJNQW
URIm2://0JWM6GJNQW
Kindstatement, proposition
WritingW0 stub
MSC15A63, 46C05
StatusThis copy of the site has no build data. Run m2 status 0JWM6GJNQW.
UsesOrthonormal family, Linear independence

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