Problem: Let $V$ be a vector space over the field with three elements. Show that there exists an endomorphism $\alpha$ of V with $\alpha(v)+\alpha(v)=v$.
I have posted a solution attempt in the answers.
Problem: Let $V$ be a vector space over the field with three elements. Show that there exists an endomorphism $\alpha$ of V with $\alpha(v)+\alpha(v)=v$.
I have posted a solution attempt in the answers.
Identify the field with three elements with $\mathbb{Z}_3$.
The condition becomes $\alpha(2v)=v$.
Fix a basis $\{b_\alpha\}$ of the vector space. We can write any $v\in V$ in the form
$v=\sum_1^n q_jb_j$
for $q_j\in \mathbb{Z}_3$. Let our map send $b_\alpha$ to $2b_\alpha$ for each basis element and extend by linearity.
Then
$\alpha(2v)=\alpha\left(\sum_1^n 2q_jb_j\right)=\sum_1^n 2\cdot2q_jb_j=\sum_1^n q_jb_j=v$
as desired.