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Unformatted text preview: part (a). Find the particular solution that satises the initial condition x = 1 3 . 4 Consider the equation for linear dynamical system x k +1 = A x k where A = 1 22 1 ! . Factorize A into r r ! cos sin sin cos ! by computing r and . Characterize the zero xed point. Explain. 5a) Show that B = { 1x, 1 + x 2 , xx 2 } forms a basis for P 2 ( R ). b) Let C = { 1 , 1 + x, 1 + x + x 2 } be another basis for P 2 ( R ). Find the change of basis matrix from B to C , using the standard basis as an intermediate basis. 6 Prove the following statements: a) Prove that det( kAI ) = k n det ( A k I ) . b) Let A and B be n n matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB = BA ....
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 Spring '08
 Yeap
 Math, Linear Algebra, Algebra

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