Page 48 - Matrix Analysis & Applied Linear Algebra

CHAPTER 2

Rectangular Systems

and

Echelon Forms

2.1 ROW ECHELON FORM AND RANK

We are now ready to analyze more general linear systems consisting of m linear
equations involving n unknowns
a 11 x 1 + a 12 x 2 + ··· + a 1n x n = b 1 ,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n = b 2 ,
.
.
.
a m1 x 1 + a m2 x 2 + ··· + a mn x n = b m ,
where m may be different from n. If we do not know for sure that m and n
are the same, then the system is said to be rectangular. The case m = n is
still allowed in the discussion—statements concerning rectangular systems also
are valid for the special case of square systems.
The first goal is to extend the Gaussian elimination technique from square
systems to completely general rectangular systems. Recall that for a square sys-
tem with a unique solution, the pivotal positions are always located along the
main diagonal—the diagonal line from the upper-left-hand corner to the lower-
right-hand corner—in the coefficient matrix A so that Gaussian elimination
results in a reduction of A to a triangular matrix, such as that illustrated
below for the case n =4:

* ∗ ∗ ∗
 
 0 * ∗ ∗ 
0 0 * ∗
T =   .
0 0 0 *