## 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 $a$th and $b$th 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 $b$th row to the $a$ th row.

It is clear that interchanging $y_a$ and $y_b$ in $S$ interchanges the $a$th and $b$th rows of $A$, since the $i$th 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 $a$th row is just the old $a$th row plus $r$ times the old $b$th row.