Consider two "set" of vectors ${\mathcal A} = \{\alpha_i \in {\mathbb R}^n\colon i = 1,\ldots,mn\}$ and ${\mathcal B} = \{\beta_i \in {\mathbb R}^m\colon i = 1,\ldots,mn\}$, where term "set" means its elements can be duplicate. Let $\gamma_i = \alpha_i \bigotimes \beta_i$, where $\bigotimes$ is the Kronecker product, and $\Gamma = \{\gamma_i = \alpha_i \bigotimes \beta_i\colon i = 1,\ldots,mn\}$.
Then, can we judge the linear dependence of $\Gamma$ by ${\mathcal A}$ and ${\mathcal B}$?
Thanks!