# calculate the maximum number of linearly independent subsets

I am not sure if this question has been asked before, apologies if I am repeating this question again.

My question is as follows:

given a set i know how to find all the subsets of the set.

But what I would really like to know is how do i find all linearly independent subsets that can be obtained from this super set.

Thanks, Bhavya

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What do you mean by linear independence? A set in itself has no linear structure, and the phrase "linear independence" is meaningless. – Willie Wong♦ Jan 24 '12 at 13:35

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@bhavya: If your set $S$ is a finite subset of a say $3$-dimensional vector space, then you need only worry about subsets of $s$ that have $3$ or fewer elements, for the other subsets are not linearly independent. Then you can test all such "small" subsets of $S$ for linear independence. Testing a single subset of $S$, say with $3$ elements, is easy, you probably have had to do it. I do not know any efficient way of testing a large number of subsets of $S$, except for the tedious process of testing one subset, then another, and so on. – André Nicolas Jan 24 '12 at 13:47

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@bhavya: As for calculating the maximum, it could easily be that all the subsets of $S$ with $3$ or fewer elements are linearly independent. So if $S$ has $n$ elements drawn from a $3$-dimensional vector space, the number of linearly independent subsets of $S$ could be as large as $\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}$. – André Nicolas Jan 24 '12 at 13:53
@Andre thank you for the explanation it makes a lot of sense – bhavya Jan 24 '12 at 15:07
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