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V. Matricer – 2. Lineære afbildninger 398 Den fuldstændige løsning er ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x1 −x2 − x3 + 1 1 −x2 −x3 ⎝x2⎠ = ⎝ x2 ⎠ = ⎝0⎠ + ⎝ x2 ⎠ + ⎝ 0 ⎠ 0 0 x3 hvor både x2 og x3 kan vælges frit. x3 ⎛ ⎞ 1 ⎛ ⎞ −1 ⎛ ⎞ −1 = ⎝0⎠ + x2 ⎝ 1 ⎠ + x3 ⎝ 0 ⎠ 0 0 1 2.25. Løsningsrummet ☞ [LA] 5 Lineære ligningssystemer Sætning 7 Løsningsmœngden til et homogent lineœrt ligningssystem med n ubekendte er et lineœrt underrum kaldet løsningsrummet eller nulrummet. Ax = 0 NA ⊂ R n Bevis Simple regneregler for matrix multiplikation giver Ax = 0, Ay = 0 ⇒ A(x + y) = 0 x3
V. Matricer – 2. Lineære afbildninger 399 2.26. 2 ligninger 4 ubkendte ☞ [LA] 5 Lineære ligningssystemer Eksempel x1 + x2 = 0 x3 + x4 = 0 (1) x3 = −x4 og x1 = −x2. (2) x4 og x2 kan vælges frit. 2.27. 2 ligninger 4 ubkendte ☞ [LA] 5 Lineære ligningssystemer Eksempel - fortsat ⎛ ⎞ x1 ⎜x2⎟ ⎜ ⎟ ⎝x3⎠ = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −x2 −1 0 ⎜ x2 ⎟ ⎜ ⎜ ⎟ ⎝−x4⎠ = x2 ⎜ 1 ⎟ ⎜ ⎟ ⎝ 0 ⎠ + x4 ⎜ 0 ⎟ ⎝−1⎠ 0 1 x4 Løsningsrummet er span af vektorerne ⎛ ⎞ −1 ⎜ 1 ⎟ ⎝ 0 ⎠ 0 , ⎛ ⎞ 0 ⎜ 0 ⎟ ⎝−1⎠ 1 x4
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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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1. Grafiske/numeriske metoder IV Di
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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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