MATH
Midterm2Solutions

# Midterm2Solutions - x = Ax A=-4-5-2-1 0 = det(I A = det 4 5...

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Unformatted text preview: Ó ÐÙ Ø iÓÒ ×ØÓ id ØeÖÑ C ÓÒ × id eÖØh e fÓ ÐÐÓÛ iÒg ×Ý ×ØeÑ Ó fd ieÖeÒ Ø ia ÐeÕÙ a Ø iÓÒ × : x prime = A x ; A = parenleftbigg- 4- 5- 2- 1 parenrightbigg a F iÒd ØÛ ÓÔ a ÖØ icÙ Ða Ö×Ó ÐÙ Ø iÓÒ × eÖ×ØÒd Øh ee igeÒÚa ÐÙ e×Ù × iÒg Øh e fÓ ÖÑ Ù Ða : 0 = det ( λI- A ) = det parenleftbigg λ + 4 5 2 λ + 1 parenrightbigg = λ 2 + 5 λ- 6 = ( λ + 6)( λ- 1) ÓØh e Öea ÖeØÛ Óe igeÒÚa ÐÙ e× λ =- 6 , 1 eÒ eÜ ØÒde igeÒÚa ÐÙ e×fÓ ÖØh e×eÙ × iÒg : ( λI- A ) vectorv = vector ÐÙgg iÒg iÒÖ×ØfÓ Ö λ =- 6 : parenleftbigg- 2 5 2- 5 parenrightbiggparenleftbigg v 1 v 2 parenrightbigg = parenleftbigg parenrightbigg →- 2 v 1 + 5 v 2 = 0 → v 1 = 5 / 2 v 2 h e Öea ÖeÑ aÒÝ iÒÒ iØe ÐÝÑ aÒÝ ÓÔ Ø iÓÒ ×h e ÖeÓÒ echÓ ice i×ØÓ×e Ø v 1 = 5 aÒd v 2 = 2 ×Ó vectorv = parenleftbigg 5 2 parenrightbigg eÖeÔ ea ØØh eabÓÚ e fÓ ÖØh e×ecÓÒd e igeÒÚa ÐÙ e λ = 1 : parenleftbigg 5 5 2 2 parenrightbiggparenleftbigg v 1 v 2 parenrightbigg = parenleftbigg parenrightbigg → 5 v 1 + 5 v 2 = 0( aÒd 2 v 1 + 2 v 2 = 0) → v 1 =- v 2 h e Öea ÖeÑ aÒÝ iÒÒ iØe ÐÝÑ aÒÝ ÓÔ Ø iÓÒ ×h e ÖeÓÒ echÓ ice i×ØÓ×e Ø v 1 = 1 aÒd v 2 =- 1 ×Ó vectorv = parenleftbigg 1- 1 parenrightbigg h eÒa Ð×Ó ÐÙ Ø iÓÒ ×haÚ e fÓ ÖÑ vectory ( t ) = vectorve λt ×ÓÔ ÐÙgg iÒg iÒÓÙ ÖØÛ Óe igeÒÚa ÐÙ e×aÒd cÓ ÖÖe×ÔÓÒd iÒg e igeÒÚ ec ØÓ Ö×Û ege ØØh eØÛ Ó×Ó ÐÙ Ø iÓÒ × : vectory 1 ( t ) = parenleftbigg 5 2 parenrightbigg e- 6 t aÒd vectory 2 ( t ) = parenleftbigg 1- 1 parenrightbigg e t b hÓÛ Øh a ØÝÓÙ Ö×Ó ÐÙ Ø iÓÒ ×a Öe ÐiÒ ea Ö ÐÝ iÒd eÔ eÒd eÒ Ø×eØh i×fac ØØÓÛ Ö iØedÓÛ Ò Øh e geÒ eÖa Ð×Ó ÐÙ Ø iÓÒ h eÒch eck iÒg fÓ Ö ÐiÒ ea Ö iÒd eÔ eÒd eÒ ceÙ ×eØh eØh eÓ ÖeÑ Øha Ø×aÝ ×a×Ó ÐÙ Ø iÓÒÓ fÚ ec ØÓ Ö×a Öe ÐiÒ ea Ö ÐÝ iÒd eÔ eÒd eÒ ØeÚ e ÖÝÛ h e Öe ifaÒdÓÒ ÐÝ ifØh eÝa Öe ÐiÒ ea Ö ÐÝ iÒd eÔ eÒd eÒ Øa ØÓÒ eÔÓ iÒ ØÓ Ü Øh eÔÓ iÒ Ø t = 0 iÒÛ h ich ca ×eØh e×Ó ÐÙ Ø iÓÒ ×a Öe : vectory 1 ( t ) = parenleftbigg 5 2 parenrightbigg aÒd vector y 2 ( t ) = parenleftbigg 1- 1 parenrightbigg h e Öea ÖeØÛ ÓÛ aÝ ×ØÓch eck fÓ Ö ÐiÒ ea Ö iÒd eÔ eÒd eÒ ceÓ fØh e×eÚ ec ØÓ Ö×Ò e i×ØÓÛ Ö iØe : c 1 vectorv 1 + c 2 vectorv 2 = 0 , aÒd ×Ó ÐÚ e fÓ Ö c 1 aÒd c 2 ÝÓÙ caÒ ×Ó ÐÚ eØh i×Ù × iÒgba × ic×k iÐÐ× fÖÓÑÑ idd Ðe×chÓÓ Ða Ðgeb Öa Y ÓÙ Û iÐÐcÓÒ c ÐÙd eØha Ø c 1 aÒd c 2 a ÖeaÒd ×Ó Øh eÚ ec ØÓ Ö×a Öe iÒd eed ÐiÒ ea Ö ÐÝ iÒd eÔ eÒd eÒ Øh e Ó Øh e ÖÛ aÝ i×ØÓcÓÒ × id e Ö : det parenleftbigg 5 1 2- 1 parenrightbigg fØh i× i×eÕÙa ÐØÓÞe ÖÓØh eÚ ec ØÓ Ö×a Öe ÐiÒ ea Ö ÐÝ deÔ eÒ edeÒ Ø Øh e ÖÛ i×eØh eÝa Öe ÐiÒ ea Ö ÐÝ iÒd e Ô...
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