The following definition is for the row reduced echelon form of a matrix.
Definition 4.3.1 Let e_{i} denote the column vector which has all zero entries except for the i^{th} slot which is one. 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 e_{1} and if you have encountered e_{1},e_{2},
For example, here are some matrices which are in row reduced echelon form.

Theorem 4.3.2 Let A be an m × n matrix. Then A has a row reduced echelon form determined by a simple process.
Proof: Viewing the columns of A from left to right take the first nonzero column. Pick a nonzero entry in this column and switch the row containing this entry with the top row of A. Now divide this new top row by the value of this nonzero entry to get a 1 in this position and then use row operations to make all entries below this entry equal to zero. Thus the first nonzero column is now e_{1}. Denote the resulting matrix by A_{1}. Consider the submatrix of A_{1} to the right of this column and below the first row. Do exactly the same thing for it that was done for A. This time the e_{1} will refer to F^{m−1}. Use this 1 and row operations to zero out every entry above it in the rows of A_{1}. Call the resulting matrix A_{2}. Thus A_{2} satisfies the conditions of the above definition up to the column just encountered. Continue this way till every column has been dealt with and the result must be in row reduced echelon form. ■
Definition 4.3.3 The first pivot column of A is the first nonzero column of A. The next pivot column is the first column after this which is not a linear combination of the columns to its left. The third pivot column is the next column after this which is not a linear combination of those columns to its left, and so forth. Thus by Lemma 4.2.3 if a pivot column occurs as the j^{th} column from the left, it follows that in the row reduced echelon form there will be one of the e_{k} as the j^{th} column.
There are three choices for row operations at each step in the above theorem. A natural question is whether the same row reduced echelon matrix always results in the end from following the above algorithm applied in any way. The next corollary says this is the case.
Definition 4.3.4 Two matrices are said to be row equivalent if one can be obtained from the other by a sequence of row operations.
Since every row operation can be obtained by multiplication on the left by an elementary matrix and since each of these elementary matrices has an inverse which is also an elementary matrix, it follows that row equivalence is a similarity relation. Thus one can classify matrices according to which similarity class they are in. Later in the book, another more profound way of classifying matrices will be presented.
It has been shown above that every matrix is row equivalent to one which is in row reduced echelon form. Note

so to say two column vectors are equal is to say they are the same linear combination of the special vectors e_{j}.
Thus the row reduced echelon form is completely determined by the positions of columns which are not linear combinations of preceding columns (These become the e_{i} vectors in the row reduced echelon form.) and the scalars which are used in the linear combinations of these special pivot columns to obtain the other columns. All of these considerations pertain only to linear relations between the columns of the matrix, which by Lemma 4.2.3 are all preserved. Therefore, there is only one row reduced echelon form for any given matrix. The proof of the following corollary is just a more careful exposition of this simple idea.
Corollary 4.3.5 The row reduced echelon form is unique. That is if B,C are two matrices in row reduced echelon form and both are row equivalent to A, then B = C.
Proof: Suppose B and C are both row reduced echelon forms for the matrix A. Then they clearly have the same zero columns since row operations leave zero columns unchanged. If B has the sequence e_{1},e_{2},
The above corollary shows that you can determine whether two matrices are row equivalent by simply checking their row reduced echelon forms. The matrices are row equivalent if and only if they have the same row reduced echelon form.
The following corollary follows.
Corollary 4.3.6 Let A be an m × n matrix and let R denote the row reduced echelon form obtained from A by row operations. Then there exists a sequence of elementary matrices, E_{1},

Proof: This follows from the fact that row operations are equivalent to multiplication on the left by an elementary matrix. ■
Corollary 4.3.7 Let A be an invertible n × n matrix. Then A equals a finite product of elementary matrices.
Proof: Since A^{−1} is given to exist, it follows A must have rank n because by Theorem 3.3.18 det(A)≠0 which says the determinant rank and hence the column rank of A is n and so the row reduced echelon form of A is I because the columns of A form a linearly independent set. Therefore, by Corollary 4.3.6 there is a sequence of elementary matrices, E_{1},

But now multiply on the left on both sides by E_{p}^{−1} then by E_{p−1}^{−1} and then by E_{p−2}^{−1} etc. until you get

and by Theorem 4.1.6 each of these in this product is an elementary matrix. ■
Corollary 4.3.8 The rank of a matrix equals the number of nonzero pivot columns. Furthermore, every column is contained in the span of the pivot columns.
Proof: Write the row reduced echelon form for the matrix. From Corollary 4.2.4 this row reduced matrix has the same rank as the original matrix. Deleting all the zero rows and all the columns in the row reduced echelon form which do not correspond to a pivot column, yields an r × r identity submatrix in which r is the number of pivot columns. Thus the rank is at least r.
From Lemma 4.2.3 every column of A is a linear combination of the pivot columns since this is true by definition for the row reduced echelon form. Therefore, the rank is no more than r. ■
Here is a fundamental observation related to the above.
Corollary 4.3.9 Suppose A is an m×n matrix and that m < n. That is, the number of rows is less than the number of columns. Then one of the columns of A is a linear combination of the preceding columns of A.
Proof: Since m < n, not all the columns of A can be pivot columns. That is, in the row reduced echelon form say e_{i} occurs for the first time at r_{i} where r_{1} < r_{2} <
Definition 4.3.10 Let A be an m × n matrix having rank, r. Then the nullity of A is defined to be n−r. Also define ker
Observation 4.3.11 Note that ker

Recall that the dimension of the column space of a matrix equals its rank and since the column space is just A
Proof: Since ker

It follows that

and so the vector in parenthesis is in ker

Since u was arbitrary, this shows

Apply A to both sides. This yields

and so each c_{i} = 0. Then the independence of the x_{j} imply each b_{j} = 0. ■