Do you need reduced row echelon form when finding the null space?

Question

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?

Answer 1

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.

Answer 2

It is not needed to convert a matrix to its row reduced echelon form (rref) or even its row echelon form. If you can find the null space i.e., the linear combination of the columns that maps them to the zero vector just by looking at the original unreduced form, then you do not necessarily have to convert the matrix to rref. Gaussian elimination and Jordan back-substitution (methods used to convert a matrix to its rref) preserve the null space of the matrix, so it does not matter if you identify the null space using the original matrix or its rref.

We convert a matrix to its rref because its easier (most of the times) to identify the null space using the rref than using the original form.