Page 56 - Matrix Analysis & Applied Linear Algebra

2.2 Reduced Row Echelon Form 49

E Notation
A
For a matrix A, the symbol E A will hereafter denote the unique re-
duced row echelon form derived from A by means of row operations.

Example 2.2.2

Problem: Determine E A , deduce rank (A), and identify the basic columns of
12231
 
 .
 24462 
36696
A = 
12453
Solution:
 1   1   1 
2231 2 2 3 1 2 2 3 1
 2 4462   0 0 00   0 0 22 
2
0
3 6696 0 0 0 0 3 0 0 0 0 3
  −→   −→  
1 2453 0 0 2 2 2 0 0 0 0 0
2 2 3 1 2 0 1 −1
1
   
1
 0 0 11   0 0 1 1 
1
1
0 0 0 0 3 0 0 0 0
−→   −→  
3
0 0 0 0 0 0 0 0 0 0
1
   
2 0 1 −1 2 0 1 0
1
 0 0 1 1   0 0 1 0 
1
1
0 0 0 0 0 0 0 0
−→   −→  
1
1
0 0 0 0 0 0 0 0 0 0
Therefore, rank (A)=3, and {A ∗1 , A ∗3 , A ∗5 } are the three basic columns.
The above example illustrates another important feature of E A and ex-
plains why the basic columns are indeed “basic.” Each nonbasic column is ex-
pressible as a combination of basic columns. In Example 2.2.2,
and A ∗4 = A ∗1 + A ∗3 . (2.2.1)
A ∗2 =2A ∗1
Notice that exactly the same set of relationships hold in E A . That is,
E ∗2 =2E ∗1 and E ∗4 = E ∗1 + E ∗3 . (2.2.2)
This is no coincidence—it’s characteristic of what happens in general. There’s
more to observe. The relationships between the nonbasic and basic columns in a