Math 2.0

Linear and multilinear algebra; matrix theory Definition

Reflection

Builds on Orthogonal projection

Definition

Let VV be an inner product space over ℝ\R or ℂ\C, and let WW be a finite-dimensional subspace of VV, with orthogonal projection PWP_W. The reflection in WW is the linear map RW ⁣:V→V,RW(v)=2PW(v)−v.R_W \colon V \to V, \qquad R_W(v) = 2 P_W(v) - v.

Remarks

RWR_W fixes each vector of WW, negates each vector of W⊥W^\perp, and satisfies RW2=id⁡VR_W^2 = \id_V. It is a linear isometry. For a hyperplane W=a⊥W = a^\perp with a≠0a \ne 0, the formula is RW(v)=v−2⟨v,a⟩⟨a,a⟩aR_W(v) = v - 2 \frac{\langle v, a \rangle}{\langle a, a \rangle} a. In ℝn\R^n, the matrix of a hyperplane reflection is orthogonal with determinant −1-1.

Reading path

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionReflection, definition, no evidence levelReflectiondefinition
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.
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  1. Vector space definition
  2. Inner product definition
  3. Linear map definition
  4. Linear subspace definition
  5. Orthogonal vectors definition
  6. Endomorphism definition
  7. Sum of subspaces definition
  8. Orthogonal complement definition
  9. Direct sum of subspaces definition
  10. A space is the direct sum of a subspace and its orthogonal complement statement
  11. Projection definition
  12. Orthogonal projection definition
  13. Reflection definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Reflection_(mathematics), Reflection through a hyperplane in n dimensions
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Tag025158VEYZ
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MSC15A04, 51F15
DefinesReflection
UsesOrthogonal projection

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