Jelly, 7 bytes

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.)

Try it online!

How?

ŒPḊ^/ÞḢ - Link: list of distinct integers
ŒP - power-set
 Ḋ - dequeue
 Þ - sort by:
 / - reduce by:
 ^ - XOR
 Ḣ - head

Jelly, 7 bytes

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.)

Try it online!

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]

Jelly, 7 bytes

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.)

Try it online!

Jelly, 7 bytes

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.)

Try it online!

How?

ŒPḊ^/ÞḢ - Link: list of distinct integers
ŒP - power-set
 Ḋ - dequeue
 Þ - sort by:
 / - reduce by:
 ^ - XOR
 Ḣ - head