The is an example:
x1 + x2 - x4 = 0
x1 + x3 - x5 = 0
The solutions are: (1,0,0,1,1), (0,1,1,0,0), (0,0,0,1,1). We know that there are 3 such solution, because there are 2 LI equations where there should be 5, and 5-2 = 3. We can find them by hand.
But how does a computer find these 3 solutions algorithmically? Or more generally:
How does a computer find all linearly independent solution of Ax = 0 ? Assuming there's always at least one solution.
Thank you.
The one that is taught most often in linear algebra classes is the process of Gaussian Elimation.
It works by converting the system of equations into matrix form and then by performing a series of elementary row operations. It can be done by hand and a variant of it is useful for finding matrix inverses if they exist.