Linear dependences over the field with two elements
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ŒPḊ^/ÞḢ


A monadic Link accepting the list of distinct integers representing the vectors which yields a subset representing the linear dependence. (May be employed as a dyadic Link also accepting the number of dimensions on the right.)

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How?

ŒPḊ^/ÞḢ - Link: list of distinct integers   e.g. [1,3,4,7]
ŒP      - power-set                              [[],[1],[3],[4],[7],[1,3],[1,4],[1,7],[3,4],[3,7],[4,7],[1,3,4],[1,3,7],[1,4,7],[3,4,7],[1,3,4,7]],[3,4,7],[1,3,4,7]]
  Ḋ     - dequeue                                [[1],[3],[4],[7],[1,3],[1,4],[1,7],[3,4],[3,7],[4,7],[1,3,4],[1,3,7],[1,4,7],[3,4,7],[1,3,4,7]],[3,4,7],[1,3,4,7]]
     Þ  - sort by:
    /   -   reduce by:
   ^    -     XOR                              ( [1,3,4,7,2,5,6,7,4,3,6,5,2,0,1] )
        - }                                      [[3,4,7],[1],[1,3,4,7],[1,3],[1,4,7],[3],[4,7],[4],[3,7],[1,4],[1,3,7],[1,7],[1,3,4],7,[3,4]]
      Ḣ - head                                   [3,4,7]