571_3.6 - 3.6 ROW SPACE AND COLUMN SPACE KIAM HEONG KWA p....

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Unformatted text preview: 3.6 ROW SPACE AND COLUMN SPACE KIAM HEONG KWA p. 154 Let A be the m n matrix. The row space of A is the subspace Row space of R 1 n spanned by the rows of A , while the column space of A is Column space the subspace of R m spanned by the columns of A . The rank of A , Rank denoted rank( A ) , is the dimension of the row space of A . Note that the i th row of A is given by e T i A for each i { 1 , 2 , ,m } , where { e 1 , e 2 , , e m } is the standard basis of R m . Hence the row space of A is also given by Span( e T 1 A, e T 2 A, , e T m A ) = ( m X i =1 c i e T i A R 1 n c i R for i = 1 , 2 , ,m ) . p. 154 Let A and B be m n matrices. If A is row equivalent to B , then Theorem 3.6.1 the row spaces of A and B are equal. Proof. It suffices to show that if B is obtained from A through a single elementary row operation, then their row spaces coincide. To begin with, say A R i R j- B , so that e T k B = e T i A if k = j, e T j A if k = i, e T k A if k 6 = i,j for all k { 1 , 2 , ,m } . Then m X k =1 c k e T k B = m X k =1 k 6 = i,j c k e T k A + c i e T j A + c j e T i A for any scalars c k s, k = 1 , 2 , ,m . This implies that the row spaces of A and B are equal provided A R i R j- B . Next, suppose A R i R i- B for a scalar 6 = 0 . Then e T k B = ( e T i A if k = i, e T k A if k 6 = i Date : July 4, 2011. 1 2 KIAM HEONG KWA for all k { 1 , 2 , ,m } , from which it follows that m X k =1 c k e T k B = m X k =1 k 6 = i c k e T k A + c i e T i A for any scalars c k s, k = 1 , 2 , ,m . This implies that the row spaces of A and B are equal provided A R i R i- B with 6 = 0 . Finally, suppose A R i R i + R j- B for a scalar , where i 6 = j . Then e T k B = ( e T i A + e T j A if k = i, e T k A if k 6 = i for all k { 1 , 2 , ,m } , from which it follows that m X k =1 c k e T k B = m X k =1 k 6 = j c k e T k A + ( c j + c i ) e T j A for any scalars c k s, k = 1 , 2 , ,m . This implies that the row spaces of A and B are equal provided A R i R i + R j- B with i 6 = j . In view of this theorem, the nonzero rows of the row echelon form of an m n matrix A form a basis of the row space of A . This is so since the nonzero rows of the row echelon form are linearly independent. In consequence of this, the number of nonzero rows of the row echelon form equals rank( A ) . p. 155 (Consistency theorem for linear systems.) Let A be an m n matrix and let b R m . The linear system A x = b is consistent if and only Theorem 3.6.2 if b is in the column space of A . In consequence of this theorem, the system A x = b is consistent for every b R m if and only if the columns of A span R m ....
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571_3.6 - 3.6 ROW SPACE AND COLUMN SPACE KIAM HEONG KWA p....

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