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VII. Skalarprodukt og projektion – 1. Ortogonal projektion 488 1.20. Projektion på basis ☞ [LA] 12 Ortogonal projektion Eksempel Lad u1 = (1, −1,1),u2 = (1,2,1) ∈ R 3 være inbyrdes ortogonale vektorer der udspænder underrummet U. Så er den ortogonale projektion projU(v) = proju1 (v) + proju2 (v) = v · u1 v · u2 u1 + u1 · u1 = v1 − v2 + v3 3 = ( v1 + v3 2 u2 u2 · u2 (1, −1,1) + v1 + 2v2 + v3 6 ,v2, v1 + v3 ) 2 (1,2,1) 1.21. Projektion på basis ☞ [LA] 12 Ortogonal projektion Eksempel 3 Betragt u1 = (1, 1 2 ,0, −1),u2 = (2,2, −1,3) ∈ R 4 samt underrummet U = span(u1,u2). 1. Vektorerne u1 og u2 er ortogonale: u1 · u2 = 2 + 1 · 2 + 0 − 3 = 0 2
VII. Skalarprodukt og projektion – 1. Ortogonal projektion 489 1.22. Projektion på basis ☞ [LA] 12 Ortogonal projektion Eksempel 3 - fortsat Betragt u1 = (1, 1 2 ,0, −1),u2 = (2,2, −1,3) ∈ R 4 samt underrummet U = span(u1,u2). 2. Lad v = (2,2,8, −6) og beregn projU(v) = proju1 (v) + proju2 (v) = v · u1 v · u2 u1 + = 9 9 4 u1 · u1 u2 u2 · u2 (1, 1 −18 ,0, −1) + (2,2, −1,3) 2 18 = (2,0,1, −7) 1.23. Pythagoras ☞ [LA] 12 Ortogonal projektion Sætning 18 (Pythagoras) Hvis a ⊥ b, så er a 2 + b 2 = a + b 2 Bevis a + b 2 = (a + b) · (a + b) = a · a + 2a · b + b · b
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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