note1 - Math 340(row space column space null space of a...

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Math 340 (row space, column space, null space of a matrix) Let A = [ a ij ] be an m by n matrix. Then 1. the row space of A is the subspace of R n spanned by the rows of A , so all vectors of the form y T A where y is in R m ; 2. the column space of A is the subspace of R m spanned by the columns of A , so all vectors of the form Ax where x is in R n ; 3. the null space of A is the subspace of R n consisting of the solutions of the homogeneous system Ax = 0. Doing EROs is the same as multiplying on the left by an invertible (i.e., nonsingular matrix) E : the matrix EA is obtained from A by doing EROs on A . The eFect of EROs on an m by n matrix A , that is, the eFect on A by multiplying A on the left by an invertible matrix, is thus: 1. ERO’s don’t change the row space of A , because y T A = y T E - 1 EA = ( y T E - 1 )( EA ) = z T ( EA ) where z T = y T E - 1 , but they can change the linear relations among the rows (so that rows that were linearly independent become dependent after EROs and vice-versa). 2. ERO’s don’t change the linear relations among the columns of
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This note was uploaded on 08/11/2008 for the course MATH 340 taught by Professor Meyer during the Spring '08 term at University of Wisconsin.

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note1 - Math 340(row space column space null space of a...

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