Kompendium til lineÃ¦r Algebra - bennike.org

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GLP 9� ± RJ GLP 9� $G L Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike 9L VNDO EORW YLVH DW DOOH NRPELQDWLRQHU DI EDVLVYHNWRUHU IRU GH WR HJHQUXP HU LQGE\UGHV RUWRJRQDOH 'HWWH J¡UHV EORW YHG DW YLVH DW VNDODUSURGXNWHUQH JLYHU ⎛−1⎞ ⎛0⎞ ⎛−1⎞ ⎛1 ⎞ ⎛1 ⎞ ⎛0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟⋅ ⎜1 ⎟ = ⎜ 0 ⎟⋅ ⎜0 ⎟ = ⎜0 ⎟⋅ ⎜1 ⎟ = 0 ⎜ 1 ⎟ ⎜0⎟ ⎜ 1 ⎟ ⎜1 ⎟ ⎜1 ⎟ ⎜0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ $OWHUQDWLYW GYV VYDUHW HU MD 3nVWDQGHQ I¡OJHU DI DW $ HU V\PPHWULVN MI NRPSHQGLHW V $G LL 9L VNDO EORW YLVH DW YHNWRUV WWHW EHVWnHQGH DI GH WUH EDVLVYHNWRUHU IUD GH HJHQUXP HU OLQH UW XDIK QJLJW IRU GHUPHG YLO GH WUH EDVLVYHNWRUHU MR GDQQH HQ EDVLV IRU R � GLP R � 2J GHWWH HU WLOI OGHW WKL −1 0 1 0 1 0 1 0 1 $OWHUQDWLYW −1 1 = 1⋅ = −2 ≠ 0 (UJR HU VYDUHW MD 1 1 3nVWDQGHQ I¡OJHU VWUDNV DI DW $ HU V\PPHWULVN MI NRPSHQGLHW V 6SHNWUDOV WQLQJHQ $G LLL 'D GHW XQGHU LL HU YLVW DW HJHQUXPPHQH VXPPHU WLO R � � � − PDQJOHU YL EDUH DW YLVH DW 9 2 9 6 & = 0 %HYLV ( ) ( ) {} [ 9� ± 9� ! [ 9� ± š [ 9� ! ⎛−1⎞ ⎛0 ⎞ ⎛1 ⎞ ⎛−1⎞ ⎛0 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∃λ 1, λ2 , λ3 ∈ 5 : [ = λ1 ⋅⎜ 0 ⎟ ∧ [ = λ2 ⋅⎜ 1⎟ + λ3 ⋅⎜ 0⎟ λ1 ⋅⎜ 0 ⎟ = λ2 ⋅⎜ 1⎟ + λ3 ⋅⎜ 0⎟ ⎜ 1 ⎟ ⎜0 ⎟ ⎜1 ⎟ ⎜ 1 ⎟ ⎜0 ⎟ ⎜1 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛−1⎞ ⎛0 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ & λ1 ⋅⎜ 0 ⎟ − λ2 ⋅⎜ 1⎟ − λ3 ⋅⎜ 0⎟ = 0 ⎜ 1 ⎟ ⎜0 ⎟ ⎜1 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ - 43 - LGHW
Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike � � � ⎛⎛− 1⎞ ⎛0⎞ ⎛1 ⎞⎞ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ 'D V WWHW ⎜⎜ 0 ⎟, ⎜1 ⎟, ⎜0⎟ ⎟ HU OLQH UW XDIK QJLJW Vn KDU RYHQVWnHQGH OLJQLQJ NXQ pQ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎝⎝ 1 ⎠ ⎝0⎠ ⎝1 ⎠⎠ O¡VQLQJ QHPOLJ RJ GHUPHG InV 0 & [ = 6YDUHW HU GHUIRU MD $OWHUQDWLYW 3nVWDQGHQ I¡OJHU GLUHNWH DI VSHNWUDOV WQLQJHQ MI LL *¡U UHGH IRU DW PDWULFHQ $ HU GLDJRQDOLVHUEDU 0DWUL[ $ HU GLDJRQDOLVHUEDU MI VSHNWUDOV WQLQJHQ NRPSHQGLHW V %HVWHP HQ RUWRJRQDO PDWUL[ 4 4� � Vn $4 HU HQ GLDJRQDOPDWUL[ RJ DQJLY 4 4� RJ ⎛⎛ 0⎞ ⎛1 ⎞ ⎛−1⎞ ⎞ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ %HWUDJW HJHQYHNWRUEDVHQ ⎜⎜ 1⎟, ⎜0⎟, ⎜ 0 ⎟⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎝⎝ 0⎠ ⎝1 ⎠ ⎝ 1 ⎠⎠ - 44 - %HP UN DW YHNWRUHUQH L GHQQH RSJDYH WLOI OGLJYLV HU LQGE\UGHV RUWRJRQDOH IUD VWDUW DI 9DU GHWWH LNNH WLOI OGHW NDQ PDQ EUXJH *UDP 6FKPLGW RUWRQRUPDOLVHULQJ MI NRPSHQGLHW V 9L VNDO VnOHGHV L GHQQH RSJDYH NXQ QRUPHUH YHNWRUHUQH RJ InU ⎛ ⎛ 1 ⎜⎛ 0⎞ ⎞ ⎛− ⎜ 2 ⎜ ⎟ ⎟ ⎜ ⎜⎜ 1⎟, ⎜ 0 ⎟, ⎜ 0 ⎜ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 1 ⎝⎝ 0⎠ ⎝ 2 ⎠ ⎝ 2 1 2 ⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟ ⎠⎠ � Vn LGHW V¡MOHUQH L 4 ⎛0 ⎜ VRP EHNHQGW HU GH RUWRQRUPDOLVHUHGH HJHQYHNWRUHU InV 4 = ⎜1 1 2 0 1 − ⎞ 2 ⎟ 0 ⎟ 'HUYHG InV ⎜ ⎝ 0 1 2 1 2 ⎟ ⎠ ⎛6 0 0 ⎞ ⎜ ⎟ −1 LI¡OJH VSHNWUDOV WQLQJHQ DW 4 $4 = ⎜0 6 0 ⎟ 'D 4 HU RUWRJRQDO KDYHV DW ⎜ ⎟ ⎝0 0 − 2⎠ 4 −1 = 4 ⎛ 0 ⎜ 1 = ⎜ 2 ⎜ 1 ⎝ − %HVWHP u PDWULFHQ 4� � 2 1 0 0 0 ⎞ ⎟ 1 2 1 2 ⎟ ⎟ ⎠ $4 � � �
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Kompendium til lineÃ¦r Algebra - bennike.org

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