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205f05-final-solutions

# 205f05-final-solutions - A5FA5FAEXA F iÒd a ÐÐ ×Ó ÐÙ...

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Unformatted text preview: A5FA5FAEXA F iÒd a ÐÐ ×Ó ÐÙ Ø iÓÒ ×Ó fØh e×Ý ×ØeÑ ×Ó feÕÙa Ø iÓÒ × : ÔÓ iÒ Ø× a x 1 + x 2 + x 3 = 1 x 1 + 2 x 3 = 2 x 1 + 2 x 2- x 3 = 3 Ó ÐÙ Ø iÓÒ : ÓÛ ÖedÙe : ÖÓÛ iÒd ia Ø iÓÒ ×a ÐÛ aÝ ×Öe fe ÖØÓ Øh eÖÓÛ ×Ó fØh eÔ ÖeÚ iÓÙ × ×ØeÔ e ÖeÛ ea ÖeÙ × iÒg Øh eaÙgÑ eÒ ØedÑ a ØÖ iÜÓ fÓÙ Ö×e 1 1 1 1 1 2 2 1 2- 1 3 ∼ 1 1 1 1- 1 1 1 1- 2 2 (1) (2)- (1) (3)- (1) ∼ 1 1 1 1 1- 1- 1- 1 3 (1)- (2) (3) + (2) ∼ 1 1 4 1- 4 1- 3 (1) + (3) (2)- (3)- (3) ∼ 1 8 1- 4 1- 3 (1)- (2) , ×Ó x 1 = 8 x 2 =- 4 aÒd x 3 =- 3 ×Ó ÐÚ e×Øh e×Ý ×ØeÑ a ×ÝÓÙaÒh ek b 2 x 1 + x 2- 8 x 3- 3 x 4 = 3 x 1- 2 x 2- 5 x 3- x 4 = 5 x 1- 8 x 2 + x 3 + 3 x 4 = A5FA5FAEXA Ó ÐÙ Ø iÓÒ : e ÖeÛ ed ea ÐÛ iØhÓÒ ÐÝ Øh eÓe ieÒ ØÑ a ØÖ iÜ : 2 1- 8- 3 3- 2- 5- 1 5- 8 1 3 ∼ 1- 10 17 9 2 1- 8- 3 3- 2- 5- 1 (3)- 2(1) (1) (2) ∼ 1- 10 17 9 21- 42- 21 28- 56- 28 (2)- 2(1) (3)- 3(1) ∼ 1- 10 17 9 1- 2- 1 (2) / 21 (3) / 28- (2) / 21 ∼ 1- 3- 1 1- 2- 1 (1) + 10(2) . h eÖaÒk i××Ó Øh ed iÑ eÒ × iÓÒÓ fØh eÒÙ ÐÐ×Ôae Øh eÒÙ ÐÐ×ÔaeÓ fØh i×Ñ a ØÖ iÜ i×Øh e×e ØÓ f×Ó ÐÙ Ø iÓÒ ×Ó fØh i××Ý ×ØeÑ i× 4- 2 = 2 h i×i×Øh e×Ôae N = Span 3 2 1 , 1 1 1 , Ó ÖØh e×e ØÓ fa ÐÐÚ eØÓ Ö× x 1 x 2 x 3 x 4 Û h e Öe x 1 = 3 x 3 + x 4 aÒd x 2 = 2 x 3 + x 4 F iÒd Øh ed e Øe ÖÑ iÒaÒ Ø×Ó fØh e fÓ ÐÐÓÛ iÒgÑ a ØÖ ie× ÔÓ iÒ Ø× a 2- 1 2 3 5 4 Ó ÐÙ Ø iÓÒ : 4 b 2 1 1 3 1 1 1 2 . Ó ÐÙ Ø iÓÒ : h eaÒ ×Û e Ö i×5Ò Øh i×ÓÒ eh ea Øedab iØA ÒÝ 2 n × 2 n Ñ a ØÖ iÜÓ f b ÐÓk fÓ ÖÑ Øha Ø i×Ñ ad eÙÔÓ f bracketleftbigg A B C D bracketrightbigg Û h e Öe A, B, C, aÒd D a Öe n × n Ñ a ØÖ ie×Û iÐÐ×a Ø i× fÝ det parenleftbiggbracketleftbigg A B C D bracketrightbiggparenrightbigg = det ( A ) det ( D )- det ( B ) det ( C ) , aÒd Øha Ø i×hÓÛÓÑ ÔÙ Øed Øh ed e Øe ÖÑ iÒaÒ ØB Ù ØÝÓÙÛ iÐÐÒd Øh e×aÑ eØh iÒg eÜÔaÒd iÒga ÐÓÒga ÖÓÛ A5FA5FAEXA F iÒd Øh e iÒÚ e Ö×eÓ fØh eÑ a ØÖ ie× : ÔÓ iÒ Ø× a bracketleftbigg 2 2 4 3 bracketrightbigg . Ó ÐÙ Ø iÓÒ : ×eØh e Ða ×× ia Ðad jÓ iÒ ØfÓ ÖÑ Ù Ða : bracketleftbigg 2 2 4 3 bracketrightbigg − 1 = 1- 2 bracketleftbigg 3- 2- 4 2 bracketrightbigg b 1 2 2 2 1 1 ....
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