Let $I = \left < f_1, \dots, f_n \right > \subset R$ be an ideal generated by homogeneous elements where $\deg(f_i) = d_i$ and $\phi$ be the graded $R$-mod homomorphism $$\phi: R(-d_1) \oplus \cdots \oplus R(-d_s) \to I$$ where $e_i \to f_i$. So it maps generators to $f_i$. If one treats $g = g_1e_1 + \dots + g_ne_n \in R(-d_1) \oplus \cdots \oplus R(-d_s) $ as a column vector $g = \begin{bmatrix} g_1\\ \vdots\\ g_n \end{bmatrix}$ with $g_i \in R(-d_i)$, then why can we view $\phi$ as $$\phi(g) = \begin{bmatrix} f_1 & \cdots &f_n \end{bmatrix} \begin{bmatrix} g_1\\ \vdots\\ g_n\end{bmatrix}.$$
I am not sure what it is asking. The map $\phi$ is a homomorphism, and it maps from vector space to vector space and is a linear map so it has a matrix representation. Is it just asking us to expand the matrix operation on the RHS of the given map ($f$ with $g$?)