
orthogonal complement
Say that you're given a basis for the null space of a matrix and you want to find a basis for the row space of that same matrix by using only the known basis for the null space. The two spaces are orthogonal complements, meaning that the dot product of any one of the basis vectors in one space with a basis vector in the other space will be zero. But actually finding a basis for the row space this way seems impossible because there are so many unknowns to solve for.

Have you done change of basis yet? If you change to a basis where the given nullspace basis looks like <(1,0,0,...), (0,1,0,...),...>, then it's trivial to write the basis for the complement. Then undo your change of basis.