Does a completely randomized matrix over a finite field have larger expected rank than other imperfect random matrices? (If this statement is vague, could you clarify this point with respect to the matrices A and B that I have given?)For example, can you compare a randomized m$\times$n matrix A compare to a following m$\times$n matrix B over $F_2$ of their expect rank?$$\begin{bmatrix} a_1 & a_2 & \cdots& a_n\\a_2 & a_3 & \cdots& a_{n+1}\\ \vdots & \vdots & & \vdots\\a_m & a_{m+1} & \cdots &a_{m+n-1}\end{bmatrix}$$which $a_i$ is randomly chosen. Intuitively, there is some correlation between the row (column) vectors of matrix B, and it seems that the row (column) vectors of A have the best independence from each other. But I don't know how to explain this intuition.
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