Reduced Row-Echelon Form
DEFINITION A row-echelon matrix in which
•each pivot is a 1 and
•each column containing a pivot contains no other nonzero entries
is said to be in reduced row-echelon form
(приведенный ступенчатый вид). |
0 1 |
|||||||||
1 |
0 |
0 |
4 |
, |
1 |
0 |
0 |
, |
||
EXAMPLES |
7 |
|
0 |
1 |
0 |
|
0 |
0 |
||
0 |
1 |
0 |
|
|
||||||
0 |
0 |
1 −1 |
|
0 |
0 |
1 |
|
0 |
0 |
|
|
0 |
|
0 |
1 |
|
0 |
0 |
0 |
11
Solving Linear Systems
EXAMPLE 1 |
0 |
0 |
5 |
→ = 5 |
0 |
1 |
0 |
−2 |
= −2 |
0 |
0 |
1 |
4 |
= 4 |
12
EXAMPLE 2 |
|
1 |
0 |
0 |
4 −1 |
|
|
|
||||||
|
|
|
|
|
|
|
0 |
1 |
0 |
2 |
6 |
|
|
|
|
1 |
|
|
+ 4 4 |
= |
0 |
0 |
1 |
3 |
2 |
−1 −4 4 |
|
||
|
|
|
−1 |
→ |
|
1 |
= |
|
||||||
|
2 |
+ 2 4 |
= |
6 |
|
2 |
= |
6 −2 4 |
|
|||||
• |
|
x |
1, |
23, |
+ 3 34 |
= |
2 |
|
|
3 |
= |
2 −3 4 |
|
|
|
|
|
x |
and x are pivots (базисныепеременные) |
|
|||||||||
• x4 is a free variable (свободнаяпеременная) |
|
|
||||||||||||
1 |
= −1 − 4, 2 = 6 −2, 3 = 2 −3, 4 = |
|
|
|||||||||||
General solution: |
= |
−1 − 4, 6 −2, 2 −3, , |
|
|||||||||||
Basis solution: |
|
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
|
or |
|
|
|
|
= 0 → |
= (−1, 6, 2, 0). |
|
. |
||||||
|
|
|
|
|
|
|
|
|||||||
13
EXAMPLE 3 |
1 |
0 |
0 0 |
|
|
0 |
1 |
2 |
0 |
|
0 |
0 |
0 |
1 |
This |
0 ∙ 1 +0 ∙ 2 + 0 ∙ 3 |
= 1 |
|
|
The last equation is |
|
|
|
|
|
|
|
. |
|
|
equation cannot be satisfied |
there is no |
||
solution to the system. |
|
|
||
The system is inconsistent!
14