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V. Matricer – 3. Lineære ligninger 412 Det reducerede ligningssystem x1 = 2 x2 + 2x3 = 16 x4 = −2 3.20. En fri tre bundne ☞ [LA] 6 Løsningsteknik Eksempel’ - igen fortsat Løsning x4 = −2 På matrix form ⎛ hvor x3 vælges frit. ⎜ ⎝ x1 x2 x3 x4 x2 = 16 − 2x3 x1 = 2 ⎞ ⎟ ⎠ = ⎛ ⎞ 2 ⎜ ⎜16 − 2x3 ⎟ ⎝ ⎠ = ⎛ ⎞ 2 ⎜ ⎜16 ⎟ ⎝ 0 ⎠ −2 x3 −2 + x3 ⎛ ⎞ 0 ⎜ ⎜−2 ⎟ ⎝ 1 ⎠ 0 3.21. Enten-eller ☞ [LA] 6 Løsningsteknik Enten-eller-princip (22) En kvadratisk matrix kan ved rækkeoperationer • enten føres over i identitetsmatricen
V. Matricer – 3. Lineære ligninger 413 • eller føres over i en matrix med en nulrække nederst Bevis Matricen på reduceret trappeform ⎛ ⎜ ⎝ 1 ? 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3.22. Struktur er sagen ☞ [LA] 6 Løsningsteknik Sætning 9 Et homogent ligningssystem, hvor der er flere ubekendte end ligninger, har altid uendelig mange løsninger. Bevis Koefficientmatricen har flere søjler end rækker. Den reducerede matrix ⎛ ⎞ 0 0 1 ? ? 0 ? ? 0 ⎜ 0 0 0 0 0 1 ? ? 0 ⎟ ⎝ 0 0 0 0 0 0 0 0 1 ⎠ 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎠ har mindst 1 pivotfri søjle. Altså er der parametre i løsningen.
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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