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How can I compute the probability of linear independency of m vectors under finite field? The vectors length assumed to be equal to n and vector elements to be random uniformly distributed. Thank you very much.

Treasure B
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  • Start by considering the choice of the first vector. If it is zero, then the second vector is not independent. So it becomes a counting problem, how many selections of $m$ linearly independent vectors can be formed. – hardmath Dec 12 '18 at 15:43
  • @hardmath "how many selections of m linearly independent vectors can be formed". Excuse me, I don't understand this. I'd like to check independency of all m vectors. I need it for getting probability that rank of matrix with these vectors is m. – Treasure B Dec 12 '18 at 16:00
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    This is just a counting exercise. You are to calculate the number of ways of selecting $m$ vectors such that they form the rows of a full rank matrix. Observe that the full rank condition is violated if and only if at some point the next vector is a linear combination of the preceding ones. How many choices does that leave for the first vector? How many for the second? At each point it is easier to calculate the number of vectors in violation. Observe that each point you can assume the earlier picks to be linearly independent (because otherwise you are already guaranteed to fail). – Jyrki Lahtonen Dec 12 '18 at 16:10
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    Essentially this is a duplicate of this. I won't cast the first vote because A) the case is not 100% clear cut, B) my vote would be binding because of my dupehammer privilege. – Jyrki Lahtonen Dec 12 '18 at 16:13
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    An important aspect of the problem setup is left out, namely the size of the finite field from which the vector entries are drawn. That must be a power of a prime, so $q=p^k$ will serve the purpose. Proceed to count the ways to choose the first vector, and continue by induction. – hardmath Dec 12 '18 at 22:17

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