After havin read this great blogpost https://codeforces.com/blog/entry/68953. I was left a bit confused. In a lot of the problems, it is assumed that any vector that is a linear combination of the basis can also be represented by a linear combination of the vectors from the vector space? Is this true? If so how to prove it. If not, why does the solution of 2a work?
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Notes that any vector in the basis is acquired by combine some original vectors.