Linear Independence of a Finite Set of Dilations by a One-Parameter Matrix Lie Group

Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally related. We study conditions for which the set ${f(e^{t_{1}A}.) ,.,f(e^{t_{m}A}.)}$ is linearly dependent in $L^{p}(\mathbb{R}^{n})$ with $1\leq p<\infty.$