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Unformatted text preview: part (a). Find the particular solution that satisﬁes 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 θsin θ sin θ cos θ ! by computing r and θ . Characterize the zero ﬁxed 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( kAλI ) = k n det ( Aλ 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, Row echelon form, elementary row operations, General vector space

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