sample_final_2_answ

Sample_final_2_answ - SAMPLE FINAL 2 ANSWERS NOTE These are only answers to the problems and not full solutions On the nal exam you will be

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SAMPLE FINAL 2 ANSWERS NOTE: These are only answers to the problems and not full solutions! On the final exam you will be expected to show all steps used to obtain your answer. 1. a) If the rank is 4 and there are 4 linear equations, then there is a leading 1 in each row of the coefficient matrix and hence the system is consistent for all ~ b . The number of parameters in the general solution is 5-4=1. b) ~v = 1( x 2 + 1) + 2( x 2 + 2 x ) + 3( x + 1) = 3 x 2 + 7 x + 4. c) ~x · ~ y = 1( - 2) + ( - 1)(1) + 2(1) = - 1, ~x × ~ y = - 3 - 5 - 1 . d) There are many, many possible choices. e) proj ~a (perp ~a ( ~x )) = ~a · ( ~x - ~x · ~a k ~a k 2 ~a ) k ~a k 2 = ~a · ~x - ~a · ~x k ~a k 2 k ~a k 2 k ~a k 2 = ~a · ~x - ~a · ~a k ~a k 2 = 0 2. Consider the system of linear equations x 1 + 2 x 2 + - x 4 = 2 - x 1 - 2 x 2 + x 3 + 4 x 4 = - 4 2 x 1 + 4 x 2 + 2 x 3 + 4 x 4 = 0 a) 1 2 0 - 1 - 1 - 2 1 4 2 4 2 4 1 2 0 - 1 2 0 0 1 3 - 2 0 0 2 6 - 4 b) x
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This note was uploaded on 12/22/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.

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Sample_final_2_answ - SAMPLE FINAL 2 ANSWERS NOTE These are only answers to the problems and not full solutions On the nal exam you will be

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