The vector space of subsets

Task number: 2507

The system of subsets of a set \(A=\{a,b,c,d,e\}\) can be viewed as a vector space over the field \(\mathbb Z_2\). Determine
  • the zero vector \(\mathbf 0\),
  • opposite vector \(-\mathbf u\) to the vector \(\mathbf u=\{b,d,e\}\),
  • the result of linear combination \(\mathbf s=1\cdot \mathbf v+1\cdot \mathbf w + 0\cdot \mathbf x + 1\cdot \mathbf y\),
    where \(\mathbf v=\{a,c,d\}\), \(\mathbf w=\{b,c\}\), \(\mathbf x=\{a,b,d,e\}\) and \(\mathbf y=\{b,e\}\),
  • whether it is possible to obtain vector \(\mathbf z=\{a,b,e\}\) as a linear combination of vectors \(\mathbf v,\mathbf w,\mathbf x\) and \(\mathbf y\).
  • Resolution

    • \(\mathbf 0=\emptyset\),
    • \(-\mathbf u=\mathbf u\), acually, this is true in any vector space over finite field of characteristic two, because \(-1=1\),
    • \(\mathbf s=\{a,d,e\}\),
    • This question leads us to the system of four equations with five variables, which in this case has no solution – so we cannot obtain the vector \(\mathbf z\) as a linear combination of \(\mathbf v,\mathbf w,\mathbf x\) and \(\mathbf y\).

    Observe, that the vectors \(\mathbf v,\mathbf w,\mathbf x\) and \(\mathbf y\) are unable to distinguish between element \(a\) and \(d\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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