TITLE: Question on finite dimensional vector space
QUESTION [5 upvotes]: Let $V$ be  a finite dimensional vector space over $F$, a finite field of two elements. Is it possible to find the sum $$\sum_{v\in V}v$$?

REPLY [2 votes]: Let $n=\dim_F V$. If $n=0$, then the sum is trivially $\mathbf0_V$, the zero vector of $V$. For $n\geq1$, consider a linear isomorphism $T:F^n\to V$. In particular your required sum is equal to $T\bigl(\sum_{w\in F^n}w\bigr)$, and the inner sum, in the case $n\geq2$, can be rewritten as
$$\sum_{v\in F^{n-1}}\bigl[(0,v)+(1,v)\bigr]=\sum_{v\in F^{n-1}}(1,2v)=\sum_{v\in F^{n-1}}\bigl(1,\mathbf0_{F_{n-1}}\bigr)=\bigl|F^{n-1}\bigr|\,\bigl(1,\mathbf0_{F_{n-1}}\bigr)=2^{n-1}\bigl(1,\mathbf0_{F_{n-1}}\bigr)=\mathbf0_{F^n}\,,$$
and so your sum is also $\mathbf0_V$ in this case. Finally, if $n=1$ then $F^1=\{0,1\}$, so your sum will be equal to the unique nonzero vector in $V$.