Ax=b after doing reduced row echelon form ie manipulation of equations using elementary algebra if we find a unique solution that means Rx=b' has a unique solution => the linear combination of columns of R gives unique points => one point in space only has one corresponding linear combination of columns of R => Ax=b most also have one unique solution as it is the same form of equations we were solving => there is a unique combination of columns of A that gives the vector b => one point in space only has one corresponding linear combination of columns of A => columns of A are linearly independent if columns of R are linearly independent?