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?



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


