Page 57 - Matrix Analysis & Applied Linear Algebra

50 Chapter 2 Rectangular Systems and Echelon Forms

general matrix A are usually obscure, but the relationships among the columns
in E A are absolutely transparent. For example, notice that the multipliers used
in the relationships (2.2.1) and (2.2.2) appear explicitly in the two nonbasic
columns in E A —they are just the nonzero entries in these nonbasic columns.
This is important because it means that E A can be used as a “map” or “key”
to discover or unlock the hidden relationships among the columns of A .
Finally, observe from Example 2.2.2 that only the basic columns to the left
of a given nonbasic column are needed in order to express the nonbasic column
as a combination of basic columns—e.g., representing A ∗2 requires only A ∗1
and not A ∗3 or A ∗5 , while representing A ∗4 requires only A ∗1 and A ∗3 .
This too is typical. For the time being, we accept the following statements to be
true. A rigorous proof is given later on p. 136.

Column Relationships in A and E A

• Each nonbasic column E ∗k in E A is a combination (a sum of mul-
tiples) of the basic columns in E A to the left of E ∗k . That is,

E ∗k = µ 1 E ∗b 1 + µ 2 E ∗b 2 + ··· + µ j E ∗b j
1 0 0 µ 1
       
 0   1   0   µ 2 
. . .
       . 
,
 .   .   .   . 
 .   .   .   . 
= µ 1   + µ 2   + ··· + µ j   =  
 0   0   1   µ j 
.
 .   .   .   
. . . .
 .   .   .   . 
0 0 0 0
’s are the basic columns to the left of E ∗k and where
where the E ∗b i
the multipliers µ i are the first j entries in E ∗k .
• The relationships that exist among the columns of A are exactly
the same as the relationships that exist among the columns of E A .
In particular, if A ∗k is a nonbasic column in A , then
, (2.2.3)
A ∗k = µ 1 A ∗b 1 + µ 2 A ∗b 2 + ··· + µ j A ∗b j
’s are the basic columns to the left of A ∗k , and
where the A ∗b i
where the multipliers µ i are as described above—the first j entries
in E ∗k .