Math 2.0

Linear and multilinear algebra; matrix theory Definition

Linear isometry

Builds on Normed vector spaceLinear map

Definition

Let VV and WW be normed vector spaces over ℝ\R or ℂ\C. A linear isometry from VV to WW is a linear map T ⁣:V→WT \colon V \to W with ∥T(v)∥=∥v∥for all v∈V.\norm{T(v)} = \norm{v} \qquad \text{for all } v \in V.

Remarks

A linear isometry is injective, because T(v)=0T(v) = 0 implies ∥v∥=0\norm{v} = 0. It keeps distances: ∥T(u)−T(v)∥=∥u−v∥\norm{T(u) - T(v)} = \norm{u - v}. Between inner product spaces, it keeps inner products, see the next statement. Between spaces of the same finite dimension, a linear isometry is bijective.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionNorm, definition, no evidence levelNormdefinitionNormed vector space, definition, no evidence levelNormed vector spacedefinitionLinear isometry, definition, no evidence levelLinear isometrydefinition
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  1. Vector space definition
  2. Linear map definition
  3. Norm definition
  4. Normed vector space definition
  5. Linear isometry definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Linear isometry
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MSC15A04, 46B04
DefinesLinear isometry
UsesNormed vector space, Linear map

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