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Unformatted text preview: a 5 = 2 a 1a 2 +3 a 4 . This allows us to read o RREF( A ) = A . 3.6.19 Let A and B be n n matrices. (a) Show that AB = 0 if and only if the column space of B is a subspace of the nullspace of A . (b) Show that if AB = 0, then the sum of the ranks of A and B cannot exceed n . Solution: (a) Partition B into columns B = [ b 1  b 2  ...  b n ]. Then AB = [ Ab 1  Ab 2  ...  Ab n ]. So if AB = 0 then we see that Ab i = 0 for all i and hence that the column space of B N ( A ). Similarly, if we had R ( B ) N ( A ), we would have Ab i = 0 for all i and hence that AB = 0. (b) If AB = 0 then from (a) we have R ( B ) N ( A ) hence rank B nullity A so that n =rank A + nullity A rank A + rank B . 1...
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 Spring '10
 Peters

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