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Unformatted text preview: A is the set of possible constant terms and is not all of R 6 , inconsistent ~ b exist. Instead of the last sentence one could observe that since there are only 5 pivots, the echelon form of A will have 65 = 1 rows that are all zero. 4. (a) Let the i th component of ~ b j be b i,j . The j th column of AB is A ~ b j = b 1 ,j ~a 1 + . . . + b n,j ~a n , a linear combination of the columns of A . Hence A ~ b j is in Col A . (b) Since every column of AB is in Col A , it follows that Col ( AB ) is a subspace of Col A . Hence dim(Col ( AB )) dim(Col A ). Since dim Col is rank, we are done. (c) We have rank ( AB ) = rank (( AB ) T ) = rank ( B T A T ). Now use (b) with A and B replaced by B T and A T respectively to obtain rank ( B T A T ) rank ( B T ). We also have rank ( B T ) = rank ( B ). Putting all this together, rank ( AB ) rank ( B )....
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 Winter '07
 Enright
 Linear Algebra, Algebra

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