Finding solution of a set of linear equations and inequalities. Set of 28 linear equations with 28

Mackenzie Rios

Mackenzie Rios

Answered question

2022-05-21

Finding solution of a set of linear equations and inequalities.
Set of 28 linear equations with 28 unknows ( x i ), so looking like this:
b i = n = 1 28 x i a i , for i = 1 . . . 28 and where a i is either 0 or 1.
Unfortunately, the rank of the A matrix (28x28), 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 x i and x i 26.
So the way I see it is that I have 17 independent equations and 2*28=56 inequalities. Is it possible to solve this problem and to find values for x i , given these 17 equations and 56 inequalities?

Answer & Explanation

dariajoq9

dariajoq9

Beginner2022-05-22Added 14 answers

Out of curiosity, why is ai either 0 or 1 and why is 1 x i 26? 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?

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?