final_spring_answ

final_spring_answ - 1. a) If the rank is 4 and there are 4...

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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 1 x 2 x 3 x 4 = 2 0 - 2 0 + - 2 1 0 0 s + 1 0 - 3 1 t. c) From our work in b) we have
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final_spring_answ - 1. a) If the rank is 4 and there are 4...

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