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Projection Theorem

Module by: Phil Schniter

Summary: This module introduces projection theorem.

  • The orthogonal projection of y onto S, where SS is a closed subspace of VV, is y ̂ =∑i<xi,y>xi y ̂ i i x i y x i s.t. xi x i is an ON basis for SS. Orthogonal projection yields the best approximation of yy in SS: y ̂ =argminx∈S∥y-x∥ y ̂ argmin x S y x The approximation error e=y- y ̂ e y y ̂ obeys to orthogonality principle: e⊥S ⊥ e S We illustrate this concept using V=ℝ3 V 3 below but stress that the same geometrical interpretation applies to any Hilbert space.
Figure 1
Figure 1 (projection.png)

A proof of the orthogonality principle is: e⊥S⇔∀i:<e,xi>=0 ⇔ ⊥ e S i e x i 0 <y- y ̂ ,xi>=0 y y ̂ x i 0 <y,xi>=< y ̂ ,xi> y x i y ̂ x i <y,xi>=<∑j<xj,y>xj,xi> y x i j j x j y x j x i <y,xi>=∑j<xj,y>¯<xj,xi> y x i j j x j y x j x i <y,xi>=∑j<y,xj> d i − j y x i j j y x j d i − j <y,xi>=<y,xi> y x i y x i

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