{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online