sam1soln

# sam1soln - MATH 43 Selected Solutions to Exam 1...

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MATH 43 Selected Solutions to Exam 1 Sample/Review October, 2007 1. Sample exam, #5 from Fall 2006: The augmented matrix has 1 - 1 - 1 2 | 1 0 0 1 - 1 | 1 0 0 0 0 | 0 as a row echelon matrix (not unique), and 1 - 1 0 1 | 2 0 0 1 - 1 | 1 0 0 0 0 | 0 as as its (unique) reduced row echelon matrix. We set x = r and z = s as parameters for the free variables, then w - x + z = 2 gives w = 2 + r - s and y - z = 1 gives y = 1 + s for the leading variables, so the solution is [ w, x, y, z ] = [2 + r - s, r, 1 + s, s ]. 2. Sample exam, #7 from Fall 2006: To decide whether ±v is in the span of ±v 1 and ±v 2 , we use the augmented matrix [ ±v 1 ±v 2 | ±v ] (see week 5 slides or Text Example 2.18, p. 90-91). We have 1 1 | 3 1 0 | 2 2 1 | 5 3 2 | 6 1 0 | 2 0 1 | 1 0 1 | 1 0 2 | 0 . We halt the row reduction, and observe that the new 2nd equation and the new 4th equation are inconsistent: c 2 = 1 and 2 c 2 = 0 . So

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## This note was uploaded on 02/29/2008 for the course MATH 43 taught by Professor Dodson during the Spring '08 term at Lehigh University .

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sam1soln - MATH 43 Selected Solutions to Exam 1...

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