  Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically
distributed random elements with law $\mu$ on the general linear group
$\textup{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n
\ldots g_1$, $n \geq 1$. Under suitable conditions on $\mu$, we establish the
first-order Edgeworth expansion for the coefficients $\langle f, G_n v \rangle$
with $v \in V$ and $f \in V^*$, in which a new additional term appears compared
to the case of vector norm $\|G_n v\|$.
