Kompendium til lineÃ¦r Algebra - bennike.org

More documents

Recommendations

Info

Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike W HU HQ HJHQY S� UGL IRU $ ! W ! 0 6 − W 0 = 0 - 41 - 2 − W ( 2 − W) ⋅( 6 − W) ⋅( 2 − W) + 0 + 0 −( 4⋅ 4⋅ ( 6 − W) + 0 + 0) = 2 2 ( 2 − W) ⋅( 6 − W) −( 16⋅ ( 6 − W ) = 0 ( 6 − W)( 2 − W) 6 − W = 0 ∨ 2 ( 2 − W) −16 = 0 W = 6 ∨ ( 2 − W) = ± 4 W = 6 ∨ ⎧− 2 W = ⎨ ⎩ 6 'YV W ^± ` � PHG UP� ± RJ UP� 4 0 0 0 ( −16) %HVWHP HJHQUXPPHW K¡UHQGH WLO VDPWOLJH HJHQY UGLHU DG W ± +HU V¡JHV [ Vn $[ ± [ 9L O¡VHU GHUIRU 1 ⎛4 0 4 0⎞ −1 * 4 ⎜ ⎟ 1 2 3 = ⎜0 8 0 0⎟ * 8 ⎜ ⎟ ⎝4 0 4 0⎠ ( $ −( − ) ( 0) (UJR KDYHV [ 1 ⎛1 ⎜ ⎜0 ⎜ ⎝0 0 1 0 1 0 0 = 0 0⎞ ⎟ 0⎟ 0 ⎟ ⎠ 4 2 − W + [ 3 = 0⎫ [ 1 = −W ⎛ [ 1 ⎞ ⎛− W ⎞ ⎛−1⎞ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ [ 2 = 0⎬ V WWHV GHUIRU W InV [ 2 ⎟ [� = 0 GYV. [ = ⎜ [ 2 = ⎜ 0 ⎟ = W ⋅⎜ 0 ⎟, W ∈R 0 = 0⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎭ [ 3 = W ⎝ [ 3 ⎠ ⎝ W ⎠ ⎝ 1 ⎠ +HUDI VHV DW ( − 9� ) DG W ⎧⎛− 1⎞⎫ ⎪⎜ ⎟⎪ 2 = VSDQ⎨⎜ 0 ⎟⎬ ⎪⎜ ⎟⎪ ⎩⎝ 1 ⎠⎭ +HU V¡JHV [ Vn $[ [ 9L O¡VHU GHUIRU ( $ − 6( 0) 3 (UJR KDYHV ⎛− 4 ⎜ = ⎜ 0 ⎜ ⎝ 4 0 0 0 4 0 − 4 0⎞ ⎟ 0⎟ 0 ⎟ ⎠ 1 * − 1 4 ⎛1 0 −1 0⎞ ⎜ ⎟ ⎜0 0 0 0⎟ ⎜ ⎟ ⎝0 0 0 0⎠
[ 1 − [ 3 = 0⎫ ⎪ 0 = 0⎬ 0 = 0⎪ ⎭ Kompendium til Lineæ r Algebra Politstudiet, matematik, 1. årsprøve Af Erik Bennike V WWHV GHUIRU [� W� RJ [� W� InV ⎛ [ 1 ⎞ ⎛W 2 ⎞ ⎛0 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ [ = ⎜ [ 2 ⎟ = ⎜ W1 ⎟ = W1 ⋅⎜ 1⎟ + W2 ⋅⎜ 0⎟, W1, W2 ∈R ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ [ 3 ⎠ ⎝W 2 ⎠ ⎝0 ⎠ ⎝1 ⎠ +HUDI VHV DW ( 9� ) ⎧⎛0 ⎞ ⎛1 ⎞⎫ ⎪⎜ ⎟ ⎜ ⎟⎪ 6 = VSDQ⎨⎜1 ⎟, ⎜0 ⎟⎬ ⎪⎜ ⎟ ⎜ ⎟⎪ ⎩⎝0 ⎠ ⎝1 ⎠⎭ %HVWHP VDPWOLJH HJHQY UGLHUV HJHQY UGLPXOWLSOLFLWHW -I VS¡UJVPnO VHV VWUDNV DW HP� ± RJ HP� $OWHUQDWLYW 'D $ HU V\PPHWULVN HU HP� UP� RJ GLVVH HU IXQGHW L VS¡UJVPnO WLO DW Y UH GH RYHQVWnHQGH MI L ¡YULJW NRPSHQGLHW V %HVWHP IRU VDPWOLJH HJHQY UGLHU HQ EDVLV IRU GH WLOK¡UHQGH HJHQUXP ⎛⎛ −1⎞⎞ ⎜⎜ ⎟⎟ (Q EDVLV IRU 9� ± HU nEHQEDUW V WWHW ⎜⎜ 0 ⎟⎟ ⎜⎜ ⎟⎟ ⎝⎝ 1 ⎠⎠ ⎛⎛ 0⎞ ⎛1 ⎞⎞ ⎜⎜ ⎟ ⎜ ⎟⎟ 9� (Q EDVLV IRU HU nEHQEDUW V WWHW ⎜⎜ 1⎟, ⎜0 ⎟⎟ ⎜⎜ ⎟ ⎜ ⎟⎟ ⎝⎝ 0⎠ ⎝1 ⎠⎠ %HVWHP GLPHQVLRQHQ DI GH L VS¡UJVPnO IXQGQH XQGHUUXP RJ XQGHUV¡J 9� 9� 9� 9� L 8QGHUV¡J RP ± A LL 8QGHUV¡J RP ± R � 9� 9� LLL 8QGHUV¡J RP ± † R � � � ⎧⎛− 1⎞⎫ ⎧⎛0 ⎞ ⎛1 ⎞⎫ ⎪⎜ ⎟⎪ ⎪⎜ ⎟ ⎜ ⎟⎪ 'D 9 ( − 2) = VSDQ⎨⎜ 0 ⎟⎬ RJ 9 ( 6) = VSDQ⎨⎜1 ⎟, ⎜0 ⎟⎬ VHV VWUDNV DW ⎪⎜ ⎟⎪ ⎪⎜ ⎟ ⎜ ⎟⎪ ⎩⎝ 1 ⎠⎭ ⎩⎝0 ⎠ ⎝1 ⎠⎭ - 42 -
Page 2 and 3: Kompendium til Lineær Algebra Poli
Page 4 and 5: Sæ tn.: 5HJQHUHJOHU IRU QRUPHU DI
Page 6 and 7: Kompendium til Lineæ r Algebra Pol
Page 8 and 9: Def.: 7UDQVSRQHULQJ DI PDWUL[SURGXN
Page 10 and 11: Sæ tn.: ,QLWLDOHWWDO RJ NRQVLVWHQV
Page 12 and 13: Sæ tn.: ,QYHUWHULQJ DI GLDJRQDOPDW
Page 14 and 15: Def.: %DVLV IRU HW XQGHUUXP Kompend
Page 18 and 19: Sæ tn.: 6XUMHNWLYLWHW RJ ELOOHGUXP
Page 22 and 23: Sæ tn.: /LQH UH DIELOGQLQJHU RJ PD
Page 28 and 29: Sæ tn.: 'HWHUPLQDQWHU DI RSHUDWLRQ
Page 30 and 31: D D Kompendium til Lineæ r Algebra
Page 34 and 35: ' = 4 -1 $4 Kompendium til Lineæ r
Page 36 and 37: Def.: .YDGUDWLVN IRUP RJ LVRWURS YH
Page 44 and 45: GLP 9� ± RJ GLP 9� $G L Kompen
Page 48 and 49: [ 9� ⎛0 ⎜ ⎝1 1 0 0 1 � 0

Kompendium til lineÃ¦r Algebra - bennike.org

Create successful ePaper yourself

Delete template?

Save as template?