I am doing some dimensional analysis from an Applied Mathematics textbook.
They present a simple 4 x 6 matrix, which gives rise to the following equations:
$a_1 - a_5 = 0$
$a_2 + 2a_5 - 3a_6 = 0$
$a_3 - a_6 = 0$
$a_4 + a_6 = 0$
The text then states that, through 'conventional linear algebra methods', the following solutions are obtained:
$a_1 = -\frac{1}{2}, a_2 = 1, a_3 = a_4 = 0, a_5 = -\frac{1}{2}, a_6 = 0$
and
$a_1 = \frac{3}{2}, a_2 = 0, a_3 = 1, a_4 = -1, a_5 = \frac{3}{2}, a_6 = 1$
I don't see the logic in moving from the homogeneous system to the given sets of linearly independent solutions. Is this a trick I haven't been taught? Is it a matter of assigning arbitrary values to the free variables $a_5$ and $a_6$?
Any help is enormously appreciated.
William