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TUE7Quiz6 - v k w k 1 w k 2 ·· w n is a basis for R n c...

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Linear Algebra Class Tue 7 Quiz 6 ID: Name: 1 Indicate whether the statement is true( T ) or false( F ). Justify your answer. (a) If A and B are n × n matrices with the same row space, then A and B have the same column space. ( F ) A = 1 0 0 0 ! , B = 1 0 1 0 ! (b) If u and v are column vectors in R n , then the rank A = uv T is 1. ( F ) u = 0 A = 0 2 Find the value(s) of λ for which the matrix A = 3 1 1 4 λ 4 10 1 1 7 17 3 2 2 4 3 has lowest rank. If λ 6 = 0, then a row echelon form of A is 1 0 0 0 0 2 0 13 0 0 2 - 5 0 0 0 0 and so rank ( A ) = 3. If λ = 0, then a row echelon form of A is 4 0 - 2 5 0 4 10 1 0 0 0 0 0 0 0 0 and so rank ( A ) = 2. Thus λ = 0 is the value for which the the matrix A has lowest rank. 3 If V = { v 1 , v 2 , ··· , v k } is a linearly independent set of vectors in R n , and if W = { w k +1 , w k +2 , ··· , w n } is a basis for the null space of the matrix A that has the vectors v 1 , v 2 , ··· , v k as its successive rows, then V W = { v 1 , v 2 , ···
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Unformatted text preview: , v k , w k +1 , w k +2 , ··· , w n } is a basis for R n . c 1 v 1 + c 2 v 2 + ··· + c k v k + c k +1 w k +1 + c k +2 w k +2 + ··· + c n w n = 0 ⇒ c 1 v 1 + c 2 v 2 + ··· + c k v k =-c k +1 w k +1-c k +2 w k +2- ··· -c n w n and span ( W ) ⊥ span ( V ) ⇒ span ( W ) ∩ span ( V ) = 0 ⇒ c 1 v 1 + c 2 v 2 + ··· + c k v k =-c k +1 w k +1-c k +2 w k +2- ··· -c n w n = 0 ⇒ c i = 0 for 1 ≤ i ≤ n ⇒ V ∪ W is a linearly independent set. dim ( span ( W ∩ V )) = n-k + k = n (= dimR n ) and span ( W ∪ V ) ⊂ R n ⇒ R n = span ( W ∪ V ) Nov. 13, 2007 Typeset by L A T E X...
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