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Let $$A =\begin{bmatrix} 1&3&-6&2\\5&12&-24&7\\9&15&-30&6\end{bmatrix}$$

$$RREF(A) =\begin{bmatrix} 1&0&0&-1\\0&1&-2&-1\\0&0&0&0\end{bmatrix}$$ $$REF(A) =\begin{bmatrix} 1&3&-6&2\\0&1&-2&1\\0&0&0&0\\\end{bmatrix}$$

Now, provided this information, it can be found that the basis for the nullspace of $A$ using the $RREF(A)$ is: $$\begin{pmatrix} \begin{bmatrix} 0\\2\\1\\0 \end{bmatrix}, \begin{bmatrix} 1\\-1\\0\\1 \end{bmatrix}\end{pmatrix}$$

Using the $REF(A)$ the "basis" for the nullspace is:

$$\begin{pmatrix} \begin{bmatrix} 6\\2\\1\\0 \end{bmatrix}, \begin{bmatrix} -2\\-1\\0\\1 \end{bmatrix}\end{pmatrix}$$

Is the "basis" that I provided for the $REF(A)$ actually a basis? How does the $REF$ and $RREF$ of a matrix differ?

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  • $\begingroup$ Difference between REF and RREF: REF: 1. Each nonzero row lies above every zero row. 2. The leading entry of a nonzero row lies in a column to the right of the column with the leading entry of any preceding row. 3. If a column contains the leading entry of some row, then all entries of that column below the leading entry are 0. RREF: the same conditions but also 4. If a column contains the leading entry of some row, then all the other entries of that column are 0. 5. The leading entry of each nonzero row is 1. Source: $\endgroup$
    – Inazuma
    Mar 29 '16 at 0:52
  • $\begingroup$ Additional note that RREF takes longer but the values can be read straight off, whereas REF requires back substitution. $\endgroup$
    – Inazuma
    Mar 29 '16 at 0:54
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The main difference is that it is easy to read the null space off the RREF, but it takes more work for the REF.

Applying a row operation to $A$ amounts to left-multiplying $A$ by an elementary matrix $E$. This preserves the null space, as $Av = 0 \iff EA v = 0$ (elementary matrices are invertible). Hence both $A$ and its RREF (and REF) have the same null space, and it is a simple matter to read off the null space from the RREF.

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From Williams (source), pg. 348:

The difference between a reduced echelon form and an echelon form is that the elements above and below a leading 1 are zero in a reduced echelon form, while only the elements below the leading 1 need be zero in an echelon form.

Examples and further discussion are given in the above text.

Another great resource is available here.

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Row Echelon form(REF) requires backward substitution while Row Reduce Echelon form (RREF) requires no backward substitution.

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