Linear Algebra, Theory And Applications, 2012a

4.3. THE ROW REDUCED ECHELON FORM 113
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. Thistimethee 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.
The following diagram illustrates the above procedure. Say the matrix looked something
like the following.
⎛
⎞
0 ∗ ∗ ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗ ∗ ∗
⎜
⎟
⎝ . . . . . . . ⎠
0 ∗ ∗ ∗ ∗ ∗ ∗
First step would yield something like
⎛
⎞
0 1 ∗ ∗ ∗ ∗ ∗
0 0 ∗ ∗ ∗ ∗ ∗
⎜
⎟
⎝ . . . . . . . ⎠
0 0 ∗ ∗ ∗ ∗ ∗
For the second step you look at the lower right corner as described,
⎛
⎞
∗ ∗ ∗ ∗ ∗
⎜
⎟
⎝ . . . . . ⎠
∗ ∗ ∗ ∗ ∗
and if the first column consists of all zeros but the next one is not all zeros, you would get
something like this.
⎛
⎞
0 1 ∗ ∗ ∗
⎜
⎟
⎝ . . . . . ⎠
0 0 ∗ ∗ ∗
Thus, after zeroing out the term in the top row above the 1, you get the following for the
next step in the computation of the row reduced echelon form for the original matrix.
⎛
⎞
0 1 ∗ 0 ∗ ∗ ∗
0 0 0 1 ∗ ∗ ∗
⎜
⎝
.
.
.
.
.
.
⎟
. ⎠ .
0 0 0 0 ∗ ∗ ∗
Next you look at the lower right matrix below the top two rows and to the right of the first
four columns and repeat the process.
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