  Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, where
$(g_n)_{n\geq 1}$ is a sequence of independent and identically distributed
random elements with law $\mu$ on the general linear group ${\rm GL}(V)$ with
$V=\mathbb R^d$. Under suitable conditions on $\mu$, we establish Cram\'{e}r
type moderate deviation expansions and local limit theorems with moderate
deviations for the coefficients $\langle f, G_n v \rangle$, where $v \in V$ and
$f \in V^*$. Our approach is based on the H\"older regularity of the invariant
measure of the Markov chain $G_n \!\cdot \! x = \mathbb R G_n v$ on the
projective space of $V$ with the starting point $x = \mathbb R v$, under the
changed measure.
