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# M2 - Math 375 Midterm Exam II Note There are five problems...

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Math 375 – Midterm Exam II October 27, 2005 Note: There are five problems. Do as much as you can during this exam. Take this exam sheet home. If you do not have time to finish all the problems you should complete the exam at home and hand in what you did for partial credit (this is due on Monday in class). 1. (i) (10pts.) Let V be the vector space of all vectors in R 3 which are linear combinations of 1 1 1 and 1 0 0 . Find an orthonormal basis of V . (ii) (15 pts) Let V be as in (i) and let a = 1 1 - 2 . Find the vector b in V which minimizes the distance x - a among all vectors x in V ; i.e. find b so that b - a = min x V x - a . Compute this minimal distance. 2. (30 pts) Consider the linear transformation R 3 R 4 given by T ( x ) = x 1 - x 2 x 2 - x 3 x 1 - x 3 0 (i) Find a basis of the nullspace of T . (ii) Find a basis of the range of T . (iii) For each b R 4 give a complete description of the set of solutions of the linear system T ( x ) = b.

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M2 - Math 375 Midterm Exam II Note There are five problems...

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