
1-44 Mathematical Methods
Note 1
A homogeneous system always has a
solution (i.e., consistent) since
r([A|B]) = r([A|O]) = r(A)
If r = n the system has trivial solution (0, 0,
0, … 0) only
If r < n the system has nontrivial solution
(i) If m = n, a nontrivial solution exists ⇔ A is
singular.
(ii) If m < n, a nontrivial solution exists; r of
the unknowns can be expressed as a linear
combination of the remaining (n − r)
unknowns to which we may assign arbitrary
values. Hence the system has an infinite
number of solutions, out of which (n − r)
are linearly independent. The rest can be
expressed as a linear combination of them.
1.16 ...