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V. Matricer – 2. Lineære afbildninger 388 2.6. Lineær afbildning til matrix ☞ [LA] 3 Lineære funktioner Sætning 5 Enhver lineœr afbildning f : R n → R m fremkommer fra en entydig bestemt m × n-matrix Bemærk A = Matr(f) f(u) = Au f(ej) = a •j “j-te søjle i matricen for f er billedet af j-te enhedsvektor i R n .” 2.7. Opgave ☞ [LA] 3 Lineære funktioner Opgave Find Matr(f) for den lineære afbildningen f(x,y) = (y,x + y). Løsning Søjlerne i Matr(f) er 1 0 0 f(e1) = f( ) = , f(e2) = f( ) = 0 1 1 Heraf 0 1 Matr(f) = 1 1 Prøve 0 1 x 1 1 y = y x + y 1 1
V. Matricer – 2. Lineære afbildninger 389 2.8. Multiplicere = sammensætte ☞ [LA] 3 Lineære funktioner Sætning 6 Lad f,g vœre lineœre afbildninger −−−−→ Rm Så er den sammensatte afbildning g ◦ f lineœr og R n f g −−−−→ R p Matr(g ◦ f) = Matr(g)Matr(f) Bevis For f(u) = Au, g(v) = Bv giver den associative lov g ◦ f(u) = g(f(u)) = B(Au) = (BA)u 2.9. Test matrix-afbildning ☞ [LA] 3 Lineære funktioner Test Den⎛lineære afbildning ⎞ ⎛f(x,y) = ⎞(x + y,x − y,y) har tilhørende matrix Matr(f): 1 1 1 1 (a) ⎝1 −1⎠. (b) ⎝1 −1⎠. 1 1 0 (c) . 1 −1 1 0 0 0 1
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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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