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Exam1oldSol - Math 441 Exam 1 x1 x2 1(8 pts a Given a...

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Math 441 Exam 1 1. (8 pts) a) Given a vector x u0304 u003d x 1 x 2 u22ee x n find A so that Ax u0304 u003d x 2 u2212 x 1 x 3 u2212 x 2 u22ee x n u2212 x n u2212 1 . (Describe the form and state the dimensions of the matrix A ) We have Ax u0304 u003d u2212 1 1 0 u22ef 0 0 u2212 1 1 0 u22f1 0 u22ee u22f1 u22f1 u22f1 0 0 u22ef 0 u2212 1 1 x 1 x 2 u22ee u22ee x n with A the indicated ue0a2 n u2212 1 ue0a3 u00d7 n matrix. b) Express the vector x 2 u2212 x 1 x 3 u2212 x 2 u22ee x n u2212 x n u2212 1 as a linear combination x 2 u2212 x 1 x 3 u2212 x 2 u22ee x n u2212 x n u2212 1 u003d x 1 ū 1 u002b x 2 ū 2 u002b ... u002b x n ū n (find the vectors ū 1 , ū 2 ,.., ū n ) We have x 2 u2212 x 1 x 3 u2212 x 2 u22ee u22ee x n u2212 x n u2212 1 u003d x 1 u2212 1 0 u22ee u22ee 0 u002b x 2 1 u2212 1 0 u22ee 0 u002b ... u002b x n u2212 1 0 u22ee u22ee 1 u2212 1 u002b x n 0 u22ee u22ee 0 1 c) What is the connection between part a) and part b)? The vectors ū 1 , ū 2 ,.., ū n are the columns of the matrix A in part a) 1
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2. (12 pts) Using Guassian elimination along with back substitution on the augmented matrix, solve the linear system Ax u0304 u003d 1 u2212 1 1 3 u2212 1 2 u2212 2 1 1 x y z u003d u2212 1 2 u2212 3 .
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