Math 2.0

statement theorem

Orthogonal projection is the closest point

Statement

Let VV be an inner product space over R\R or C\C, let WW be a finite-dimensional subspace of VV, and let v∈Vv \in V. Let PWP_W be the orthogonal projection onto WW. Then, for all w∈Ww \in W, ∥v−PW(v)∥≤∥v−w∥,\norm{v - P_W(v)} \le \norm{v - w}, with equality only for w=PW(v)w = P_W(v).

Remarks

The proof: v−w=(v−PW(v))+(PW(v)−w)v - w = (v - P_W(v)) + (P_W(v) - w) is a sum of orthogonal vectors, so the Pythagorean theorem applies. The number ∥v−PW(v)∥\norm{v - P_W(v)} is the distance from vv to WW. The Hilbert projection theorem gives the same result for closed convex sets in Hilbert spaces. This is the basis of the method of least squares.

Explanations

L1 human

The nearest point of a subspace to a given vector is the foot of the perpendicular. Every other point of the subspace is farther away, by the Pythagorean theorem. This turns problems of best approximation into problems of projection.

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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 subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal 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 projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…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.
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  1. Vector space definition
  2. Inner product definition
  3. Linear map definition
  4. Linear subspace definition
  5. Norm definition
  6. Inner product space definition
  7. Orthogonal vectors definition
  8. Endomorphism definition
  9. Sum of subspaces definition
  10. Norm induced by an inner product definition
  11. Orthogonal complement definition
  12. Direct sum of subspaces definition
  13. A space is the direct sum of a subspace and its orthogonal complement statement
  14. Projection definition
  15. Orthogonal projection definition
  16. Orthogonal projection is the closest point statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Hilbert_projection_theorem, Finite dimensional case

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