2.3 The rank of a Matrix and Application

# 2.3 The rank of a Matrix and Application - The rank of a...

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The rank of a Matrix and Application Definition . Let A = [ a 11 a 12 …….a 1 n a 21 a 22 ……a 2 n …………………… .. a i 1 a i 2 ……a ¿ ………………… .. a m 1 a m 2 …a mn ] be an m×n matrix. The rows of A , a ( ¿¿ 11 ,a 12, …,a 1 n ) v 1 = ¿ a ( ¿¿ 21 ,a 22, …,a 2 n ) v 2 = ¿ ………………………. a ( ¿¿ m 1 ,a m 2, … ,a mn ) v m = ¿ considered as vectors in R n , span a subspace of called the row space of A . Similarly, the columns of A , w 1 = [ a 11 a 21 a m 1 ] ,w 2 = [ a 12 a 22 a m 2 ] ,…,w n = [ a 1 n a 2 n a mn ]

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considered as vectors in R m , span a subspace of called the column space of A . Definition. An elementary row operation on m×n matrix A is any of the following operations: a) Interchange rows r and s of A . That is , replace a r 1 , a r 2, …,a rn by a s 1 , a s 2, …,a sn and a s 1 , a s 2, …,a sn by a r 1 , a r 2, …,a rn . b) Multiply row r of A by c≠ 0 . That is , replace a r 1 , a r 2, …,a rn by ca r 1 , ca r 2, …,ca rn . c) Add d times row r of A to row s of A ,r≠ s. That is , replace a s 1 , a s 2, …,a sn by a s 1 + d a r 1 ,a s 2 + d a r 2 ,…,a sn + d a rn . An important notice: The elementary row operations do not change the row space of the matrix. Equivalence Whenever B can be derived from A by a combination of elementary row and column operations, we write A ~ B , and we say that A and B are equivalent matrices . We can say that A ~ B
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