C3093616 - 16(a(Statement of the problem Proof 1 Take x N(A...

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16 (a) (Statement of the problem.) Proof 1. Take x N ( A ). Then A x = 0 by definition of N ( A ). So ( BA ) x = B ( A x ) = B 0 = 0 . Hence x N ( BA ) by definition of null space. It means that N ( A ) N ( BA ) . (1) Take x N ( BA ). Then BA x = 0 by definition of null space. So B - 1 BA x = B - 1 0 . That is, A x = 0 . Hence x N ( A ). It means that N ( BA ) N ( A ) . (2) By (1) and (2), we have N ( BA ) = N ( A ) . Since N ( BA ) = N ( A ), dim N ( BA ) = dim N ( A ). By the Rank-Nullity Theorem, rank( A ) + dim N ( A ) = n, and rank( BA ) + dim N ( BA ) = n. So we get rank( A ) = rank( BA ), i.e. the rank of A is equal to the rank of BA . / Proof 2. Since B is nonsigular, it is a product of elementary matrices
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