I have a set of 28 linear equations with 28 unknows ($x_i$), so looking like this:
$b_i$ = $\sum_{n=1}^{28} x_i \cdot a_i$, for $i = 1 ... 28$ and where $a_i$ is either 0 or 1.
Unfortunately, the rank of the $A$ matrix (28 x 28), containing the parameters $a_{i,j}$ is 17. However, I do know for every of the 28 unknowns $x_i$ that it should be a whole number and that:
1 $\leq$ $x_i$ and $x_i$ $\leq$ 26.
So the way I see it is that I have 17 independent equations and 2 x 28 = 56 inequalities. Is it possible to solve this problem and to find values for $x_i$, given these 17 equations and 56 inequalities? If so, how do I approach this problem? (A solution does exist since I’m working on an exercise.)
Thanks.
Edit: My question is, can I use the inequalities to make a determined system that has a unique solution? Right now, the system is undetermined since I have 17 equations and 28 unknowns.
Out of curiosity, why is $a_i$ either 0 or 1 and why is $1\le x_i\le26$? Or is that just what you've been given?
In general, when the number of equations (call it $m$) is less than the number of unknowns (call it $n$), the geometric interpretation is a linear transformation from $n$ dimensions to $m$ dimensions. So for example you are trying to find the the three-dimensional vectors which land on a plane after the transformation, or the two-dimensional vectors which land on a line after the transformation. It is logical that any solutions to a constraint may not be unique; the linear transformation is not injective as distinct input vectors must map to the same output vector.
Why can't you just move forward with elimination like you would for a square matrix, and substitute in 1 or 0 to the free columns?