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Unformatted text preview: w gives v = ( A-I ) w as a desired member of the column space, as long as it is non zero. 2. You are given that: T = 1 1-2-1 ! , A = 1 2-4-5 ! , and T-1 AT =-3-1 ! . Use this information to ﬁnd a particular solution to X = AX +-1 1 ! . Hint: Use the substitution X = TY to ﬁnd the Special Fundamental Matrix for the given system and then use it. Answer: We know that the Special Fundamental Matrix is given by the formula: Φ( t ) = T e-3 t e-t ! T-1 and variation of parameters gives the answer: X = Φ( t ) u where u = Φ-1 ( t )-1 1 ! = T e 3 t e t ! T-1-1 1 ! We thus solve: u = T-e t ! and get u = T-e t ! . Thus, a particular solution is T e-3 t e-t ! T-1 T-e t ! = T-1 ! =-1 1 ! . 2...
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- Fall '09
- Linear Algebra, special fundamental matrix