32 CHAPTER 1. REVIEW OF SOME LINEAR ALGEBRA

Next take −2 times the middle row and add to the bottom row followed by multiplying themiddle row by −1 :  1 3 11 10 36

0 1 3 1 130 0 0 0 0

 .

Next take −3 times the middle row added to the top: 1 0 2 7 −30 1 3 1 130 0 0 0 0

 . (1.11)

At this point it is clear that the last column is −3 times the first column added to 13 timesthe second. By Lemma 1.6.10, the same is true of the corresponding columns in the originalmatrix A. As a check,

−3

 111

+13

 321

=

 362310

 .

You should notice that other linear relationships are also easily seen from (1.11). Forexample the fourth column is 7 times the first added to the second. This is obvious from(1.11) and Lemma 1.6.10 says the same relationship holds for A.

This is really just an extension of the technique for finding solutions to a linear system ofequations. In solving a system of equations earlier, row operations were used to exhibit thelast column of an augmented matrix as a linear combination of the preceding columns. Therow reduced echelon form just extends this by making obvious the linear relationshipsbetween every column, not just the last, and those columns preceding it. The matrix in1.11 is in row reduced echelon form. The row reduced echelon form is the topic of the nextsection.

1.7 The Row Reduced Echelon Form Of A MatrixWhen you do row operations on a matrix, there is an ultimate conclusion. It is calledthe row reduced echelon form. We show here that every matrix has such a row reducedechelon form and that this row reduced echelon form is unique. The significance is that itbecomes possible to use the definite article in referring to the row reduced echelon form.Hence important conclusions about the original matrix may be logically deduced from anexamination of its unique row reduced echelon form. First we need the following definition.

Definition 1.7.1 Define special column vectors ei as follows.

ei =(

0 · · · 1 · · · 0)T

.

Recall that T says to take the transpose. Thus ei is the column vector which has all zeroentries except for a 1 in the ith position down from the top.

Now here is the description of the row reduced echelon form.

Definition 1.7.2 An m× n matrix is said to be in row reduced echelon form if,in viewing successive columns from left to right, the first nonzero column encountered is