Page 101 - DMTH502_LINEAR_ALGEBRA

Unit 5: Summary of Row-Equivalence

From the conditions (1) one has R = 0 if i > s and j  k . Thus Notes
ij s
 = (0,..., 0,  ,...,b ), b  0
ks n ks
and the first non-zero coordinate of  occurs in column k . Note also that for each k , S = 1,..., r,
8 8
there exists a vector in W which has a non-zero k th coordinate, namely  .
s s
It is now clear that R is uniquely determined by W. The description of R in terms of W is as
follows. We consider all vectors  = (b ,...,b ) in W. If   0, then the first non-zero coordinate of
1 n
 must occur in some column t:
 = (0,...,0, b ,..., b ), b  0
t n t
Let k ,...,k be those positive integers t such that there is some   0 in W, the first non-zero
1 r
coordinate of which occurs in column t. Arrange k ,...,k in the order k < k < ... < k . For each of the
1 r 1 2 r
positive integers k there will be one and only one vector  in W such that the k th coordinate of
s s s
 is 1 and the k th coordinate of  is 0 for i  s. Then R is the m × n matrix which has row vectors
s i s
 ,..., , 0, ..., 0.
1 r
Corollary. Each m × n matrix A is row-equivalent to one and only one row-reduced echelon
matrix.
Proof: We know that A is row-equivalent to at least one row-reduced echelon matrix R. If A is
row-equivalent to another such matrix R’, then R is row-equivalent to R’; hence, R and R’ have
the same row space and must be identical.

Corollary: Let A and B be m × n matrices over the field F. Then A and B are row-equivalent if and
only if they have the same row space.
Proof: We know that if A and B are row-equivalent, then they have the same row space. So
suppose that A and B have the same row space. Now A is row-equivalent to a row-reduced
echelon matrix R and B is row-equivalent to a row-reduced echelon matrix R’. Since A and B have
the same row space, R and R’ have the same row space. Thus R = R’ and A is row-equivalent to B.
To summarize—if A and B are m × n matrices over the field F, the following statements are
equivalent:
1. A and B are row-equivalent.
2. A and B have the same row space.
3. B = PA, where P is an invertible m × m matrix.

A fourth equivalent statement is that the homogeneous systems AX = 0 and BX = 0 have the same
solutions; however, although we know that the row-equivalence of A and B implies that these
systems have the same solutions, it seems best to leave the proof of the converse until later.

5.4 Summary

 Such an equation is called a linear combination of the equations in (1). Evidently any
solution of the entire system of equations (1) will also be the solution of this new equation.
This is the fundamental idea of the elimination process.
 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).
 An m × n matrix B is called row-reduced if:
(a) the first non-zero entry in each non-zero row of B is equal to 1;

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