Midterm2Solutions

Midterm2Solutions - i id e C id eh e f ig e fd iee ia e a i...

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Unformatted text preview: i id e C id eh e f ig e fd iee ia e a i : x prime = A x ; A = parenleftbigg- 4- 5- 2- 1 parenrightbigg a F id a ic a i ed h ee igea e ig h e f a : 0 = det ( I- A ) = det parenleftbigg + 4 5 2 + 1 parenrightbigg = 2 + 5 - 6 = ( + 6)( - 1) h e ea e e igea e =- 6 , 1 e e de igea ef h ee ig : ( I- A ) vectorv = vector gg ig if =- 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 ea e a i ie a i h e e ech ice ie v 1 = 5 ad v 2 = 2 vectorv = parenleftbigg 5 2 parenrightbigg ee ea h eab e f h eecd e igea e = 1 : parenleftbigg 5 5 2 2 parenrightbiggparenleftbigg v 1 v 2 parenrightbigg = parenleftbigg parenrightbigg 5 v 1 + 5 v 2 = 0( ad 2 v 1 + 2 v 2 = 0) v 1 =- v 2 h e ea e a i ie a i h e e ech ice ie v 1 = 1 ad v 2 =- 1 vectorv = parenleftbigg 1- 1 parenrightbigg h ea i ha e f vectory ( t ) = vectorve t gg ig i e igea ead c ed ig e ige ec ege h e i : vectory 1 ( t ) = parenleftbigg 5 2 parenrightbigg e- 6 t ad vectory 2 ( t ) = parenleftbigg 1- 1 parenrightbigg e t b h h a i a e i ea id e ed e eh ifac ied h e ge ea i h ech eck ig f i ea id e ed e ce eh eh e e ha a a i f ec a e i ea id e ed e e e h e e ifad ifh ea e i ea id e ed e a e i h e i t = 0 i h ich ca eh e i a e : vectory 1 ( t ) = parenleftbigg 5 2 parenrightbigg ad vector y 2 ( t ) = parenleftbigg 1- 1 parenrightbigg h e ea e a ch eck f i ea id e ed e ce fh ee ec e i ie : c 1 vectorv 1 + c 2 vectorv 2 = 0 , ad e f c 1 ad c 2 ca eh i igba ick i f idd ech a geb a Y ic c d eha c 1 ad c 2 a ead h e ec a e id eed i ea id e ed e h e h e a ic id e : det parenleftbigg 5 1 2- 1 parenrightbigg fh i iea e h e ec a e i ea de e ede h e ieh ea e i ea id e...
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Midterm2Solutions - i id e C id eh e f ig e fd iee ia e a i...

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