Given a system of homogeneous equations such as $$x_1 - x_2 - x_3 = 0\\x_2-x_4-x_5=0\\ x_3-x_5-x_6=0,$$
I am interested in finding maximal subsets of the variables $\{x_1,\ldots,x_n\}$ such that (1) the variables in the set can be assigned values "freely", i.e. without constraint, (2) when those variables are specified, the system of equations uniquely determines the values of all other variables.
For example, in this case, $\{x_1, x_2,x_5\}$ and $\{x_4,x_5,x_6\}$ are maximal sets in this sense, while $\{x_1,x_2,x_3\}$ fails to satisfy property (1), and $\{x_1, x_4\}$ fails to satisfy property (2).
This sense of "independence" feels like it should relate to the geometric/algebraic properties of the matrix representing this system of equations, but I do not see a direct connection to anything I know about, e.g. the null space or column space of a matrix. Maybe I'm overlooking something.
Is there a straightforward way to find these sets?
Edit: Through "non-pivot" variables, Gaussian elimination can help identify a single independent set, which is a useful start. I am wondering how to enumerate all such independent sets, or test a candidate set of variables for this property, or a geometric interpretation of these sets. It could be helpful to understand pivot rows through the language of linear operators rather than matrices.
From an operator perspective, I believe that at least the number of non-pivot variables is equal to the dimension of the nullspace of the operator. I am not sure what the relationship is between the different bases of the nullspace and the independent variable sets themselves.
Maybe this is useful: A subset $I$ of variables has this property iff there is a basis $\{b_i : i\in I\}$ for the null space such that for all $i,j \in I$, we have $b_{i,j} = \begin{cases}1&i=j\\0&i\neq j\end{cases}.$