Properties of the rows of a relations matrix

Let R, M, \varphi, x_j, y_i, and A be defined as in this previous exercise.

  1. Show that interchanging the generators y_a and y_b has the effect of interchanging the ath and bth rows of the relations matrix A.
  2. Show that, for any r \in R, replacing y_a by y_a + ry_b a \neq b) gives another generating set S^\prime for \mathsf{ker}\ \varphi. Show further that the matrix with respect to this generating set is obtained from A by adding r times the bth row to the a th row.

It is clear that interchanging y_a and y_b in S interchanges the ath and bth rows of A, since the ith row of A is merely the B-expansion of y_i.

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 S^\prime = \{y_1, \ldots, y_a + ry_b, \ldots, y_b, \ldots, y_m\} is indeed a generating set for \mathsf{ker}\ \varphi. Moreover, it is clear that the new ath row is just the old ath row plus r times the old bth row.

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