Let , , , , , and be defined as in this previous exercise.
- Show that interchanging the generators and has the effect of interchanging the th and th rows of the relations matrix .
- Show that, for any , replacing by ) gives another generating set for . Show further that the matrix with respect to this generating set is obtained from by adding times the th row to the th row.
It is clear that interchanging and in interchanges the th and th rows of , since the th row of is merely the -expansion of .
Our proof in the previous exercise that adding a multiple of one element in a generating set to another element does not change the generated submodule holds here as well, so the altered set is indeed a generating set for . Moreover, it is clear that the new th row is just the old th row plus times the old th row.