# quiz-sol - -x 1 4 x 3 = 200-x 1 x 2 9 x 3 = 200 2 x 1-x 2 7...

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MATH 407 QUIZ NAME (Please print): Solutions There are 2 problems. Stop now and make sure you have both problems. If you do not have them both, then request a new quiz. The first problem is worth 30 points and the second is worth 45 points for a total of 75 points. Show all of your work and follow the directions provided. Partial credit will be given for partial solutions. CALCULATORS ARE NOT ALLOWED! Problem Score 1 2 Total

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Page 2 of 2 1. [a](15 points) Under what conditions is the set S IR n a subspace of IR n . Solution S is a subspace of IR n if (i) 0 S (ii) x + y S for all x, y S . (iii) αx S for all α IR and x S . [b](15 points) Let A be a linear transformation from IR n to IR m with m < n . Give the definition for the null space of A and provide a simple lower bound for its dimension. Solution Nul( A ) = { x IR n | Ax = 0 } and dim(Nul( A )) = n - dim(Ran( A )) n - m
2. Consider the system
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Unformatted text preview: -x 1 + 4 x 3 = 200-x 1 + x 2 + 9 x 3 = 200 2 x 1-x 2 + 7 x 3 = 200 . (a)(15 points) Write the augmented matrix corresponding to this system. Solution -1 4-1 1 9 2-1 7 ± ± ± ± ± ± ± 200 200 200 (b)(20 points) Reduce the augmented system in part (a) to echelon form. Solution-1 4 200-1 1 9 200 2-1 7 200 1-4-200-r 1 1 5 r 2-r 1-1 15 600 r 3 +2 r 1 1-4-200 1 5 20 600 r 2 + r 3 (c)(10 points) Describe the set of solutions to the given system. Solution 20 x 3 = 600 → x 3 = 30 x 2 + 5 x 3 = 0 → x 2 =-5 x 3 =-150 x 1-4 x 3 =-200 → x 1 =-200 + 4 x 3 =-80 Therefore, the solution set consists of the unique point x 1 x 2 x 3 = -80-150 30 ....
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