HW9-510-soln - AMS 510 HW9 Solution 4.83 1 2 u= 2 v = 3 w...

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AMS 510 HW9 Solution 4.83 u = 1 2 3 , v = 2 3 1 , w = mu nv (a) w = 3 u 1 –u 2 (b) Impossible (c) 1 2 1 2 3 k 3 1 4 = 0 k = 11 / 5 (d) 1 2 a 2 3 b 3 1 c = 0 7a 5b c = 0 4.84 Expand f = xp 1 yp 2 zp 3 , we can get x = a , y = 2 a b , z = a b c . 4.87 For any u span S , there exists v 1 , v 2 , v n S , such that u = a 1 v 1 a 2 v 2 ⋯ a n v n , then u = a 1 v 1 a 2 v 2 ⋯ a n v n 1 0 span S ∪{ 0 } , that is span S ⊆ span S ∪{ 0 } . For any u span S ∪{ 0 } , there exists v 1 , v 2 , v n S and 0 ∈{ 0 } , such that u = a 1 v 1 a 2 v 2 ⋯ a n v n a n 1 0 , then u = a 1 v 1 a 2 v 2 ⋯ a n v n span S , that is span S ∪{ 0 } ⊆ span S . In sum, span S = span S ∪{ 0 } 4.88 (a) For any u span S , there exists v 1 , v 2 , v n S , such that u = a 1 v 1 a 2 v 2 ⋯ a n v n , since S T , v 1 , v 2 , v n T , then
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HW9-510-soln - AMS 510 HW9 Solution 4.83 1 2 u= 2 v = 3 w...

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