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Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of our vectors will be linearly independent (over $\mathbb{F}_2$) too.

The question is as follows: how large $d - n$ should be if we want this probability to be, say, $1 - 1 / \mathrm{poly}(n)$?

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Ilya, is this your homework in probability? Sounds like an advanced problem for serious students at MSU. –  Wadim Zudilin May 4 '10 at 13:17
No. This fact would be useful for my research. Our course in probability is not that advanced. –  ilyaraz May 4 '10 at 13:22
Then please clarify whether all objects (vectors, random $d$-dimensional subspaces) are with respect to ground field $\mathbb F_2$, or the 2-element field is needed to express the independence property. Even the latter option sounds strange, a better formulation of the problem would be helpful. –  Wadim Zudilin May 4 '10 at 13:41
We consider linear dependence over $F_2$ of course. –  ilyaraz May 4 '10 at 13:54
This is clear. Are the vectors from $\mathbb F_2^{2n}$? Is the subspace viewed as a subspace of $\mathbb F_2^{2n}$? –  Wadim Zudilin May 4 '10 at 14:14

1 Answer 1

up vote 10 down vote accepted

Suppose the vectors are $e_1,\dots,e_n$. The kernel of projection onto a random subspace of dimension $n+r$ is a random subspace of dimension $n-r$, so you want the probability that such a subspace has trivial intersection with the span of $e_1,\dots, e_n$. Now just count the number of choices for a basis $v_1,\dots, v_{n-r}$ of such a space: $2^{2n} - 2^n$ for the first vector, then $2^{2n} - 2^{n+1}$ for the second, and so on. This is to be compared with $2^{2n} - 1$ choices for the first vector if one doesn't have an restriction, $2^{2n}-2$ for the second and so on.

So the probability of this happening is the ratio of these two quantities, which you need to find a good approximation for; a very brief back-of-an-envelope calculation suggested it's about $1 - c2^{-r}$, at least if $r$ is largeish. For your specific needs, then, $d - n$ should be about $C\log n$.

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