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flat-dist

# flat-dist - Distance between flats Jonathan L.F King...

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Unformatted text preview: Distance between flats Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 15 November, 2011 (at 07:53 ) For a general field F and vectors u , w in an F-vector- space, define the “ line through u in direction w” : LinDir( u , w ) := { u + t w | t ∈ F} . We now happily specialize to an IP ( inner-product ) space V over field F ⊂ C which is sealed under complex-conjugation; ∀ α ∈ C : α ∈ F = ⇒ α ∈ F . Our IP is conjugate-linear in its 1 st argument. I.e, for every β ∈ F and u , w ∈ V : h β u , w i = β · h u , w i and h u ,β w i = β · h u , w i . Proj and Orth. For a direction-vector D 6 = and arbitrary u ∈ V , we define the orthogonal-projection operator: Proj D ( u ) := h D , u i h D , D i · D . 1 : Our IP is linear in its 2 nd argument, so formula (1) indeed satisfies that Proj D ( β u ) = β · Proj D ( u ). This immediately gives that Proj D is idempotent: Write w := Proj D ( u ) = α D , where α := h D , u i h D , D i . Then Proj D ( w ) equals α · Proj D ( D ) = α D = w . Let’s also check that the difference u- Proj D ( u ) is orthogonal to D : Well, D , u- Proj D ( u ) equals h D , u i - h D , w i = h D , u i - α ·h D , D i...
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