Determine the basis of the row space of the matrix: |1 4 0| |-1 -3 3| |2 9 5|

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We want to determine the basis of the row space of the matrix:

`([1,4,0],[-1,-3,3],[2,9,5])`

One basis is found by row-reducing the matrix; the row vectors are then a basis.

`([1,4,0],[-1,-3,3],[2,9,5])->([1,4,0],[0,1,3],[0,1,5])` R2=R1+R2;R3=-2R1+R3

`->([1,4,0],[0,1,3],[0,0,-2])`  R3=-R2+R3

`->([1,4,0],[0,1,3],[0,0,1])`  Multiply R3 by `-1/2`

**Elementary row operations do not change the row space **

So the row vectors `w_1=(1,4,0),w_2=(0,1,3),w_3=(0,0,1)` form a basis.

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