Can anyone give me some basic idea how 1051 can be solved?I
think the limits are too extreme!It sounds like the
knapsack problem.I can't figure out how to check if a
system of vectors is linearly independent in reasanoble time
(I am 11th grade).I think finding the determinant for big
matrices is impossible because the complexity is n!(I want
to check if the determinant is 0).

Idea is to add on each step an lineary independent vector
with the lowest cost. Then when you have a matrix with
vectors' coordinates you need to make it triangle (like :

x1 x2 x3 x4
0  y2 y3 y4
0  0  z3 z4

and so on...)

You must keep this form when adding a vector, so you don't
need to calculate determinant, you either will get or 0-
vector (making matrix triangle) - so vector was linear
dependent, or some new vector that you'll add to matrix,
then you always can say (in O(amount_of_vector) time) is
vector linear dependent with other in matrix or not.

I think you are suggesting Gaussian elimination, I normaly
don't know the Gaussian elimination algorithm, but I did a
web search and found it, because of this problem.Anyway I
have decided not to work on the problem because I am in
highschool and I think this problem requires quite a bit
knowledge about matrices which I don't posses.
Thanks for replying and trying to help.