## The vector space of subsets

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 $$d$$.