Math 2.0

definition

Reflection

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.

Explanations

L1 human

A reflection is a mirror image in a subspace. Vectors in the mirror stay where they are, and vectors perpendicular to it change direction. Reflecting twice brings every vector back.

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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
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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

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Sources

  1. Web https://en.wikipedia.org/wiki/Reflection_(mathematics), Reflection through a hyperplane in n dimensions

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