Math 2.0

statement proposition

A linear isometry preserves the inner product

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.

Explanations

L1 human

If a linear map keeps lengths, then it also keeps angles. The reason is that lengths determine inner products, by polarisation. So rigid linear motions keep the whole geometry.

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To read this node, read these 10 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 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 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.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Isometry, Linear isometry

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