Some questions about the concept of underdetermined systems

Question

I'm reading a linear algebra textbook and I have some confusion about the concept of underdetermined systems $A\mathbf{x}=\mathbf{y}$:

• First, we know that for each vector $\mathbf{y}$ in $\mathbb{R}^m$ the underdetermined linear system is either inconsistent or has infinitely many solutions. So are there some theorems that tell us when the system is inconsistent and when it has infinitely many solutions?

• Second, if an underdetermined linear system has infinitely many solutions, is it guaranteed that a positive solution (all elements in $\mathbf{x}$ are positive) exists?

It would be very appreciated if anyone could give some explanation on them.

Answer 1

There is indeed a theorem:

The linear system $Ax=b$ has a solution if & only if the matrix $A$ and the augmented matrix $[A|b]$ have the same rank. If this is the case, the common rank is the codimension of the (affine) subspace of solutions.