Are the coefficients of a linear combination of random vectors as random?

Question

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?

Answer 1

So $A = Y X^{-1}$, where $X$ and $Y$ are the $n \times n$ matrices with these vectors as columns, and $A$ is the matrix with entries $a_{ij}$.
Almost surely, $A$ has full rank because $Y$ and $X^{-1}$ do. Thus the rows of $A$ are almost surely linearly independent, as are the columns.

Answer 2

Your conjecture that the $a_i$ are independent is incorrect.

Consider, for example, $N=2$, $x_1$ and $x_2$ and $y$ each independent uniform randoms on $(1,0)\times(0,1)$. Then, using upper indices as vector indices $$ \begin{array}{cc} x_i^1 = \pmatrix{1 \\ 0} \cdot x_i && x_i^2 = \pmatrix{0 \\ 1} \cdot x_i \\ y^1 = \pmatrix{1 \\ 0} \cdot y &&y^2 = \pmatrix{0 \\ 1} \cdot y \end{array} $$ we can express the $a_i$ as $$ a_1 = \frac{x_2^2y^1-x_1^2y^2}{x_2^2x_1^1-x_1^2x_2^1} $$ and a similar expression for $a_2$. Now write out the distribution of $a_1$, and the distribution of $a_1$ givven that $a_2$ is some fixed value. You will find those distributions are not the same (in particular, the distribution for $a_1$ is broader when $a_2$ is large), so $a_1$ and $a_2$ are not iid.