Math 2.0

Linear and multilinear algebra; matrix theory Definition

Orthogonal projection

Builds on A space is the direct sum of a subspace and its orthogonal complementProjection

Definition

Let VV be an inner product space over ℝ\R or ℂ\C, and let WW be a finite-dimensional subspace of VV. By the orthogonal decomposition, each v∈Vv \in V is v=w+w′v = w + w' with unique w∈Ww \in W and w′∈W⊥w' \in W^\perp. The orthogonal projection onto WW is the map PW ⁣:V→VP_W \colon V \to V, PW(v)=wP_W(v) = w.

Remarks

PWP_W is a linear projection with image WW and kernel W⊥W^\perp, and it is self-adjoint. For an orthonormal basis e1,…,eke_1, \dots, e_k of WW, PW(v)=∑i=1k⟨v,ei⟩eiP_W(v) = \sum_{i=1}^{k} \langle v, e_i \rangle e_i. For the line spanned by u≠0u \ne 0, P(v)=⟨v,u⟩⟨u,u⟩uP(v) = \frac{\langle v, u \rangle}{\langle u, u \rangle} u. PW(v)P_W(v) is the point of WW closest to vv.

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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 projectiondefinition
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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

Where this leads

34 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Projection_(linear_algebra), Orthogonal projections
About this page
Tag0Q8HZTJC7G
URIm2://0Q8HZTJC7G
Kinddefinition
WritingW0 stub
MSC15A04, 46C05
DefinesOrthogonal projection
UsesA space is the direct sum of a subspace and its orthogonal complement, Projection

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