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 \(d\).
