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I have heard several people say that you need to put a matrix in reduced row echelon form to find a basis for the null space of a matrix. But why does it not suffice to simply go down to row echelon form (not necessarily reduced row echelon form) and then get a basis from there?

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It is more convenient to find the linearly independent rows and columns when your matrix is in reduced echelon form. It is worth to take the extra steps to go from echelon form to reduce echelon form.

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