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# HW05 - EE 203000 Linear Algebra Homework#5(Due BEFORE class...

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EE 203000 Linear Algebra Homework #5 (Due December 25, 2009 BEFORE class) Note: Detailed derivations are required to obtain a full score for each problem. (Total 110%) 1. (6%+6%) Problem 2(f) and Problem 3(d) of Section 5.1. 2. (6%+4%+6%+6%) Let T be a linear operator on a ±nite-dimensional vector space V . (a) Let us de±ne the determinant of T as det([ T ] β ) for any ordered basis β for V . Show that the determinant of T is independent of the choice of the ordered basis. That is, show that det([ T ] β ) = det([ T ] γ ) for any two ordered basis β and γ for V . (b) Prove that T is invertible if and only if zero is not an eigenvalue of T . (c) Prove that λ is an eigenvalue of T if and only if λ 1 is an eigenvalue of T 1 . (d) Prove that the eigenspace of T corresponding to the eigenvalue λ is the same as the eigenspace of T 1 corresponding to λ 1 . 3. (8%) Find the general solution to the following system of di²erential equations x ( t ) = 2 x ( t ) + y

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HW05 - EE 203000 Linear Algebra Homework#5(Due BEFORE class...

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