Linearly independent rows in a binay matrix

Question

I need the algorithm to finding only the linearly independent rows in a binary matrix using XOR function.

Example 1:

The result:

Example 2:

The result:

R4 is not included because:

Answer 1

If you do row reduction on $A^T$ (transpose of $A$), the columns with (row) leading $1$s will give you a maximal set of linearly independent columns of $A^T$. The corresponding rows of $A$ would give you a maximal set of linearly independent rows of $A$. (The reason for the transposing is that row reduction preserves column dependencies, but not row dependencies).