Page 58 - Matrix Analysis & Applied Linear Algebra

2.2 Reduced Row Echelon Form 51
Example 2.2.3

Problem: Write each nonbasic column as a combination of basic columns in
 
2 −4 −863
A =  0 1 323  .
3 −2 008

Solution: Transform A to E A as shown below.
   3   3 
2
1
1
−4 −863 −2 −43 −2 −4 3
2 2
0 1 323 → 0 1 3 2 3 → 0 3 2 3 →
1
     
3 −2 008 3 −2 0 0 8 0 4 12 −9 7
2
 15   15   
0 2 7 0 2 7 0 2 0 4
1
1
1
2 2
1
1
1
→
→
 0 3 2 3   0 3 2 3   0 3 0 2 
0 0 0 −17 − 17 0 0 0 1 0 0 0 1
1
1
2 2 2
The third and fifth columns are nonbasic. Looking at the columns in E A reveals
1
E ∗3 =2E ∗1 +3E ∗2 and E ∗5 =4E ∗1 +2E ∗2 + E ∗4 .
2
The relationships that exist among the columns of A must be exactly the same
as those in E A , so
1
and A ∗5 =4A ∗1 +2A ∗2 + A ∗4 .
A ∗3 =2A ∗1 +3A ∗2
2
You can easily check the validity of these equations by direct calculation.
In summary, the utility of E A lies in its ability to reveal dependencies in
data stored as columns in an array A. The nonbasic columns in A represent
redundant information in the sense that this information can always be expressed
in terms of the data contained in the basic columns.
Although data compression is not the primary reason for introducing E A ,
the application to these problems is clear. For a large array of data, it may be
more efficient to store only “independent data” (i.e., the basic columns of A )
along with the nonzero multipliers µ i obtained from the nonbasic columns in
E A . Then the redundant data contained in the nonbasic columns of A can
always be reconstructed if and when it is called for.
Exercises for section 2.2
2.2.1. Determine the reduced row echelon form for each of the following matri-
ces and then express each nonbasic column in terms of the basic columns:
211 30 41
 
   424 41
1233  55 

(a)  2469  , (b)  213 10 43 
 .
 634
81

2676 95 
003 −30 03 

842 14 1133