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V. Matricer – 1. Vektorer og matricer 378 Eksempel Rækkesum ⎛ Søjlesum ⎜ ⎝ a11 ... a1n . aij . am1 ... amn ⎜ 1, ..., 1 ⎝ ⎞ ⎛ ⎞ ⎛ ⎞ 1 a11 + · · · + a1n ⎟ ⎜ . ⎠ ⎝ . ⎟ ⎜ . ⎠ = . ⎝ . ⎟ . ⎠ 1 am1 + · · · + amn ⎛ a11 ... a1n . aij . am1 ... amn ⎞ ⎟ ⎠ = a11 + · · · + am1, ..., a1n + · · · + amn 1.22. Vigtigste regneregel ☞ [LA] 2 Matricer Sætning 1 (Associativ lov) Matrix multiplikation er associativ. Givet A en m × n-matrix, B en n × p-matrix og C en p × qmatrix, så er følgende to m × q-matricer ens. Bevis (AB)C = A(BC)
V. Matricer – 1. Vektorer og matricer 379 Fælles il-te indgang dil = j,k aijbjkckl 1.23. Multiplikation og linearkombination ☞ [LA] 2 Matricer Sætning 2 Givet A en m × n-matrix og x en n-søjlematrix, så er produktet y = Ax = a •1x1 + · · · + a •nxn den m-søjlematrix, der fremkommer som linearkombinationen af søjlerne i A med koefficienter de n indgange i x. Bevis Udregn yi = j aijxj 1.24. Nemme regneregler ☞ [LA] 2 Matricer Bemærkning Simple regneregler For matricer af størrelser, så operationerne er definerede gælder
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Forord "Slides" til forelæsningern
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I. Differentiation - 1. Kontinuitet
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I. Differentiation - 2. Partielle a
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I. Differentiation - 3. Tangentplan
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I. Differentiation - 4. Kædereglen
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I. Differentiation - 6. Maksimum/mi
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I. Differentiation - 7. Lagrangemet
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II. Integration - 1. Dobbelt integr
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II. Integration - 2. Itereret integ
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II. Integration - 3. Generelle omr
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II. Integration - 4. Koordinatskift
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III. Potensrækker - 1. l’Hospita
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III. Potensrækker - 2. Talfølger
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III. Potensrækker - 3. Potensrækk
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III. Potensrækker - 4. Taylorpolyn
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1. Grafiske/numeriske metoder IV Di
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IV. Differentialligninger - 1. Graf
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IV. Differentialligninger - 2. 1. o
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V. Matricer - 4. Determinanter 429
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1. Egenvektorer VI Egenvektorer og
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VI. Egenvektorer og diagonalisering
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VII. Skalarprodukt og projektion -
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VIII. Appendiks - 1. Polære koordi
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IX. Opgaver - 1. August 2002 524 1.
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IX. Opgaver - 5. Januar 2005 594 (t
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Litteratur • Arnold, Ordinary dif
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0-række/søjle, 423 n-te afsnitssu
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Identitetsmatricen, 383 Identitetsm
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etningsafledede, 89, 102 retningsfe
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