Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A linear isometry preserves the inner product

Builds on Linear isometryPolarisation identity

Statement

Let VV and WW be inner product spaces over ℝ\R or ℂ\C, with the induced norms, and let T ⁣:V→WT \colon V \to W be a linear isometry. Then, for all u,v∈Vu, v \in V, ⟨T(u),T(v)⟩=⟨u,v⟩.\langle T(u), T(v) \rangle = \langle u, v \rangle.

Remarks

The proof applies the polarisation identity, which writes inner products in terms of norms. Conversely, a linear map that keeps inner products keeps norms. So a linear isometry also keeps angles and orthogonality.

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 map, definition, no evidence levelLinear mapdefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionNormed vector space, definition, no evidence levelNormed vector spacedefinitionLinear isometry, definition, no evidence levelLinear isometrydefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionPolarisation identity, statement, no evidence levelPolarisation identitystatementA linear isometry preserves the inner product, statement, no evidence levelA linear isometry pre…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 map definition
  4. Norm definition
  5. Inner product space definition
  6. Normed vector space definition
  7. Linear isometry definition
  8. Norm induced by an inner product definition
  9. Polarisation identity statement
  10. A linear isometry preserves the inner product statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Linear isometry
About this page
Tag0R7VXTYHYZ
URIm2://0R7VXTYHYZ
Kindstatement, proposition
WritingW0 stub
MSC15A04, 46C05
StatusThis copy of the site has no build data. Run m2 status 0R7VXTYHYZ.
UsesLinear isometry, Polarisation identity

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