Differences Between Row Echelon and Reduced Row Echelon

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?

• 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: Mar 29 '16 at 0:52
• Additional note that RREF takes longer but the values can be read straight off, whereas REF requires back substitution. Mar 29 '16 at 0:54

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.

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.

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