I have to devise an algorithm to decide which columns will take part on a linear system's square matrix, by taking one column at a time and ensuring the system has a solution in the end.
I figured that I could place the matrix in column echelon form (row echelon form of the transpose, $REF(A^T)$), one given column at a time, and I'll know if the newly added column is linearly independent from all the others previously added columns. If not, I'll reject it, and use the next column suggestion. Since $|A| = |A^T|$, rows will be linearly independent iff columns also are, giving me a system with unique solution.
Now I have to solve the system, and I already have $REF(A^T)$. Can it be used somehow to help me solve the system (I know the right hand side beforehand), or I'll have to perform Gaussian elimination all over again in order to obtain $REF(A|b)$ and solve it?