What can be said about the solutions to a homogenous system?

The statement that the solutions to a homogeneous system always include the zero vector is accurate. In linear algebra, a homogeneous system is defined by the equation (Ax = 0), where (A) is a matrix and (x) is a vector of variables. The zero vector (typically denoted as (0)) satisfies this equation, as multiplying any matrix (A) by the zero vector results in the zero vector (i.e., (A \cdot 0 = 0)).

Furthermore, when solving a homogeneous system, one can also find other solutions beyond the zero vector, depending on the dimensionality and properties of the matrix (A). However, the inclusion of the zero vector as a solution is guaranteed because homogeneous systems always have at least one solution, and that solution is the zero vector.

The other options are not correct in this context: solutions do not exclusively exclude the zero vector, cannot be limited to only positive values, and are not restricted to rational numbers; they can include a broader set of solutions depending on the parameters of the variables involved.