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# Substituting into the coefficient matrix the system

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Unformatted text preview: The fundamental matrix for the system x x is given by Direct multiplication results in ________________________________________________________________________ page 408 —————————————————————————— CHAPTER 7. —— Hence 15 . Let be arbitrary, but fixed, and variable. Similar to the argument in Prob. , the columns of the matrix are linear combinations of fundamental solutions. Hence the columns of are also solution of the system of equations. Further, setting , I That is, is a solution of the initial value problem Z AZ , with Z Now consider the change of variable . Let W Z . The given initial value problem can be reformulated as W Since that AW , with W is a fundamental matrix satisfying A , with I , it follows W That is, . Based on Part W Z . , I . Hence . It also follows that 16. Let A be a diagonal matrix, with A positive integer, , A e e e e e e . Note that for any . It follows, from basic matrix algebra, that ____________________________________...
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