Given are $n$ random vectors $x_i\in\mathbb{C}^n$ and a vector $y\in\mathbb{C}^n$ which entries (of both $x_i$ and $y$) are drawn iid from some continuous distribution. Every set of $n$ different of those vectors is almost surely linearly independent. What can be said about the vector $a=[a_1,\ldots,a_n]^\top$ which consists of the unique coefficients of the linear combination $$y = \sum_{i=1}^n a_i x_i \quad\text{?}$$ Can it be proven that $a$ is iid taken from some continuous distribution? Do you have references to the literature which say something about that?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is almost surely linearly independent. What can be said about the vector $a_i=[a_{i,1},\ldots,a_{i,n}]^\top$ which consists of the unique coefficients of the linear combination $$y_i = \sum_{j=1}^n a_{i,j} x_j \quad\text{?}$$ Are the $a_i$ for $i=1,\ldots,n$ almost surely linear independent? Do you have references to the literature which say something about that?