Kompendium til lineÃ¦r Algebra - bennike.org

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[ 9� ⎛0 ⎜ ⎝1 1 0 0 1 � 0⎞ ⎟ 0⎟ ⎠ ! Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike ⎛0 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎜ ⎟ 0[ 1 + 1[ 2 + 0[ 3 = 0⎫ [ ⊥⎜1 ⎟ ∧ [ ⊥⎜0 ⎟ ⎬ 9L EHWUDJWHU GHUIRU ⎜ ⎟ ⎜ ⎟ 1[ 1 + 0[ 2 + 1[ 3 = 0⎭ ⎝0 ⎠ ⎝1 ⎠ ⎛1 ⎜ ⎝0 ⎛ [ 1 ⎞ ⎛− W ⎞ ⎜ ⎟ ⎜ ⎟ [ = ⎜ [ 2 ⎟ = ⎜ 0 ⎟ � Vn 9 ⎜ ⎟ ⎜ ⎟ ⎝ [ 3 ⎠ ⎝ W ⎠ $G 9� ± � [ 9� ± � ⎛ [ 1 ⎞ ⎛ V⎞ ⎜ ⎟ ⎜ ⎟ [ = ⎜ [ 2 ⎟ = ⎜ W ⎟ � Vn 9 ⎜ ⎟ ⎜ ⎟ ⎝ [ 3 ⎠ ⎝ V⎠ 0 1 ( 6) 1 0 ⊥ 0⎞ ⎟ 0⎟ ⎠ 6 WWHV GHUIRU [� W InV ⎧ ⎛−1⎞ ⎫ ⎧⎛−1⎞⎫ ⎪ ⎜ ⎟ ⎪ ⎪⎜ ⎟⎪ = ⎨[ [ = W ⋅⎜ 0 ⎟, W ∈ 5⎬ = VSDQ⎨⎜ 0 ⎟⎬ ⎪ ⎜ ⎟ ⎪ ⎪⎜ ⎟⎪ ⎩ ⎝ 1 ⎠ ⎭ ⎩⎝ 1 ⎠⎭ ⎛−1⎞ ⎜ ⎟ ! [ ⊥ ⎜ 0 ⎟ − [ 1 + [ 3 = [� 0 6 [� WWHV GHUIRU W RJ V InV ⎜ 1 ⎟ ⎝ ⎠ ( − 2) 28) 9LV DW XQGHUUXPPHW 8 VSDQ ⊥ ⎧ ⎛0 ⎞ ⎛1 ⎞ ⎫ ⎧⎛0 ⎞ ⎛1 ⎞⎫ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪⎜ ⎟ ⎜ ⎟⎪ = ⎨[ [ = W ⋅⎜ 1⎟ + V ⋅⎜ 0⎟, W ∈ 5⎬ = VSDQ⎨⎜1 ⎟, ⎜0 ⎟⎬ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪⎜ ⎟ ⎜ ⎟⎪ ⎩ ⎝0 ⎠ ⎝1 ⎠ ⎭ ⎩⎝0 ⎠ ⎝1 ⎠⎭ ⎧⎛ − 3⎞⎫ ⎪⎜ ⎟⎪ ⎨⎜ 6 ⎟⎬ ⎪⎜ ⎟⎪ ⎩⎝ − 3⎠⎭ ⊆ 9� IUD VS¡UJVPnO 9� � 'D HU HW XQGHUUXP HU GHW QRN DW YLVH DW 6 ∈9 ( 6) ⎛− 3⎞ ⎛0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 6 ⎟ = 6⋅ ⎜1 ⎟ + ⎜ ⎟ ⎜ ⎟ ⎝− 3⎠ ⎝0 ⎠ ( − 3) ⎛1 ⎞ ⎜ ⎟ ⋅⎜ 0⎟ ⎜ ⎟ ⎝1 ⎠ - 47 - ⎛− 3⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 3⎟ ⎝− ⎠ RJ GHWWH HU NODUW LGHW /DG PDWUL[ $ Y UH GHQ WLO HQ NYDGUDWLVN IRUP 4 K¡UHQGH PDWUL[ %HVWHP $V GHILQLWKHGVIRUKROG ,I¡OJH VS¡UJVPnO HU $V HJHQY UGLHU RJ ± 'HUIRU HU 4 HOOHU $ LQGHILQLW MI NRPSHQGLHW V
DG $ EHWHJQH +HVVHPDWULFHQ � IXQNWLRQ" Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike IRU HQ IXQNWLRQ I R � o R .DQ I Y UH HQ NRQYHNV HOOHU NRQNDY 'D +HVVHPDWULFHQ HU LQGHILQLW HU GHQ VnOHGHV KYHUNHQ SRVLWLY VHPLGHILQLW HOOHU QHJDWLY VHPLGHILQLW RJ NDQ GHUIRU KYHUNHQ Y UH NRQYHNV HOOHU NRQNDY -I 0$ V RJ 0$ V %HWUDJW HQ OLQH U DIELOGQLQJ 6 R � o R IRU KYLONHQ GHW J OGHU DW 6 H� H� H� 6 H� H� 6 H� H� H� KYRU H� H� H� EHWHJQHU GHQ NDQRQLVNH EDVLV IRU R � L %HJUXQG NRUW DW H� LNNH HU HQ HJHQYHNWRU IRU 6 H� H� H� LL %HVWHP GHQ WLO 6 K¡UHQGH PDWUL[ $ PKW LLL %HJUXQG DW 6 HU VXUMHNWLY $G L 'HWWH HU NODUW WKL 6 H� H� H� N Â H� N R (UJR HU H� LNNH HQ HJHQYHNWRU IRU 6 $G LL 'D V¡MOHYHNWRUHUQH L GHQ WLOK¡UHQGH PDWUL[ $ HU ELOOHGHUQH DI GH NDQRQLVNH EDVLVYHNWRUHU InV OHW $G LLL ⎛2 ⎜ $ = ⎜0 ⎜ ⎝4 0 6 0 4⎞ ⎟ 0⎟ 2⎟ ⎠ %LOOHGUXPPHW IRU $ XGVS QGHV DI GH OLQH UW XDIK QJLJH V¡MOHYHNWRUHU L $ 'D UJ $ LIOJ VS¡UJVPnO Vn KDYHV DW 5 $ R � (UJR HU 6 VXUMHNWLY %HWUDJW HQ OLQH U DIELOGQLQJ 7 R � o R � ⎛1 ⎞ ⎛2 ⎞ ⎜ ⎟ ⎜ ⎟ 7⎜ 1⎟ = ⎜6 ⎟, ⎜ ⎟ ⎜ ⎟ ⎝0 ⎠ ⎝4 ⎠ ⎛1 ⎞ ⎛6 ⎞ ⎜ ⎟ ⎜ ⎟ 7⎜ 0⎟ = ⎜0 ⎟, ⎜ ⎟ ⎜ ⎟ ⎝1 ⎠ ⎝6 ⎠ ⎛0 ⎞ ⎛8 ⎞ ⎜ ⎟ ⎜ ⎟ 7⎜ 1⎟ = ⎜6 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝2 ⎠ ⎝4 ⎠ 1 Se MA1, s. 395 for en definition af Hessematricen. IRU KYLONHQ GHW J OGHU DW - 48 -
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Kompendium til lineÃ¦r Algebra - bennike.org

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