Midterm2Solutions - Ó ÐÙ Ø iÓÒ ×ØÓ id ØeÖÑ C...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern