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V. Matricer – 2. Lineære afbildninger 386 Bemærk Det er nok, at sum og skalarmultiplikation bevares f(u + v) = f(u) + f(v) f(αu) = αf(u) 2.3. Lineær afbildning ☞ [LA] 3 Lineære funktioner Eksempel Afbildningen f : R 2 → R 2 givet ved er lineær. Bevis Tilsvarende for skalarmultiplikation. f(x,y) = (y,x + y) f((x1,y1) + (x2,y2)) = f(x1 + x2,y1 + y2) = (y1 + y2,x1 + x2 + y1 + y2) = (y1,x1 + y1) + (y2,x2 + y2) = f(x1,y1) + f(x2,y2) 2.4. Matrix til lineær afbildning ☞ [LA] 3 Lineære funktioner Definition
V. Matricer – 2. Lineære afbildninger 387 For en m × n-matrix A defineres en afbildning ved Eksempel u1 ↦→ u2 1 2 3 4 R n → R m u ↦→ Au u1 u2 u1 + 2u2 = 3u1 + 4u2 2.5. Matrix til lineær afbildning ☞ [LA] 3 Lineære funktioner Sætning 4 Funktionen f : R n → R m f(u) = Au defineret ved en m × n-matrix A er lineœr Bevis f(u + v) = f(u) + f(v) f(αu) = αf(u) A(u + v) = Au + Av, A(αu) = αAu Fra de simple regneregler for matrix multiplikation.
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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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Litteratur • Arnold, Ordinary dif
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0-række/søjle, 423 n-te afsnitssu
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