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So I have a linearly independent subset of $\mathbb{F}_q^n$ and I'm trying to extend it to a basis. I heard this can be done, but I can't find an algorithm for actually doing it. Can you help me?

Also, what if it wasn't a subset of $\mathbb{F}_q^n$, but rather an arbitrary vector space $V$?

Thanks!

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    @DavidMitra The set of vectors of length $n$ with elements in the field $\mathbb{F}_q$2012-04-06

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The gaussian elimination algorithm might be useful. Take a look at here. If you put your vectors as rows of a matrix and you compute the echelon form you should add the vector $e_i$ if there is no row starting (with $1$) at column $i$.

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    @JyrkiLahtonen Cool observation,2012-04-06