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Here $S=\mathbb Q[x,y]$, and we define $\oplus Se_i$ to be a $S$-free module with basis $\{e_1,e_2,e_3\}$. Define a map from $\oplus Se_i$ to $S$ by $e_1\to x^2$, $e_2\to xy+y^2$, $e_3\to y^3$. Is the representation matrix of this map $$\begin{pmatrix} x^2 & xy+y^2 & y^3 \\ \end{pmatrix}.$$ I am not sure and feel confused, could you give me some ideas? Thank you!

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You have a map $\phi:S^3\to S$ given by $\phi(e_1)=x^2$ and so on. (In this case $e_1=(1, 0, 0)$ and so on.) The matrix of this map is (as in the vector spaces case) the following: $(\phi(e_1)\ \phi(e_2)\ \phi(e_3))$, so your guess is correct.

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