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I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k.

For example:

k = 4, and therefore I have 2^k = 16 vectors to select from.

I want to randomly select 7 vectors (no repetition).

What is the probability that 4 of the selected vectors forms a linearly independent set (4 x 4 matrix with rank 4)

and

the other 3 forms a linearly independent set (4 x 3 matrix of rank 3)? These two sets are viewed independently from each other.

I can calculate the probability of obtaining a single set of linearly independent vectors, but I cannot figure out how to calculate 2 sets at the same time.

Any help would be greatly apprectiated.

Thank YOU

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# Multiple sets of linearly independent vectors

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