Page 93 - DMTH502_LINEAR_ALGEBRA

Unit 5: Summary of Row-Equivalence

Let us say that two systems of linear equations are equivalent if each equation in each system is Notes
a linear combination of the equations in the other system. We then formally state the following
theorem:
Theorem 1: Equivalent systems of linear equations have exactly the same solutions.
Consider now the system (1) as given by the system (2). We call A the matrix of coefficients of the
system. We wish now to consider operations on the rows of the matrix A which correspond to
forming linear combinations of the equations in the system AX = Y. We restrict ourselves to
three elementary row operations on m × n matrix A over the field F:
1. Multiplication of one row of A by a non-zero scalar c;
th
2. Replacement of the r row of A by row r plus c times row s, c is any scalar and r  s;
3. interchange of two rows of A.

An elementary row operation is thus a special type of function (rule) e which is associated with
each m × n matrix (A). One can precisely describe e in the three cases as follows:
1. e(A) = A if i  r, e (A) = cA
ij ij rj rj
2. e(A) = A if i  r, e(A) + cA
ij ij rj rj
3. e(A) = A if i is different from r and s, e(A) = A ,
ij ij rj si
e(A) = A
sj rj
A particular e is defined on the class of all m rowed matrices over F. One reason that we restrict
ourselves to these simple types of row operations is that having performed such an operation e
on a matrix A, we can recapture A by performing a similar operation on e(A).

Definition: If A and B are m × n matrices over the field F, we say that B is row-equivalent to A if
B is obtained from A by a finite sequence of elementary row operations. Consider the two
systems of equations
AX = 0,
and BX = 0.
If matrix B is obtained from A by a finite sequence of elementary row operations we say that B
matrix is row equivalent to A. Hence the above two system of equations are equivalent and so
they have the same solutions.

Example 1: Consider
AX = 0

  1 i

where A   i 3 
 
 
 1 2
so the system of equations is
–x + ix = 0
1 2
–ix + 3x = 0
1 2
x + 2x = 0
1 2

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