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Unformatted text preview: H~x = ~ 0 as before. Example. Page 140 number 4. Theorem 2.5. Rank Equation. Let A be m n with row-echelon form H . (1) The dimension of the nullspace of A is nullity( A ) = (# free variables in solution of A~x = ~ 0) = (# pivot-free columns of H ) . (2) rank( A ) = (# of pivots in H ) . (3) Rank Equation: rank( A ) + nullity( A ) = # of columns of A. 2.2 The Rank of a Matrix 3 Theorem 2.6. An Invertibility Criterion. An n n matrix A is invertible if and only if rank( A ) = n. Example. Page 141 number 12. If A is square, then nullity( A ) = nullity( A T ). Proof. The column space of A is the same as the row space of A T , so rank( A ) = rank( A T ) and since the number of columns of A equals the number of columns of A T , then by the Rank Equation: rank( A ) + nullity( A ) = rank( A T ) + nullity( A T ) and the result follows. QED...
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- Fall '04