sample_final_2_answ

# sample_final_2_answ - SAMPLE FINAL 2 ANSWERS NOTE These are...

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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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sample_final_2_answ - SAMPLE FINAL 2 ANSWERS NOTE These are...

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