Let $k$ be a field with five elements. Let $V$ be the $k-$vector space of $5\times1$ matrices with entries in k. Let $S$ be a subset of $V$ such that $u^{t}v =0$ for all $u$ and $v$ in $S$ here $u^t$ denotes the transpose of $u$ and $u^{t}v$ the usual matrix product. What is the maximum possible cardinality of $S? $
solution i tried- Given that set $S$ has elements of the from of $5\times 1$ matrix let
$$u=\begin{pmatrix} a\\ b\\ c\\ d\\ e\end{pmatrix} and \;\;v=\begin{pmatrix} f\\ g\\ h\\ i\\ j\end{pmatrix} $$ now according to question in set $S$ given that $u^tv=0$ i.e $$af+bg+ch+id+ej=0$$ or we can say that $$af+bg+ch+id+ej$$ is multiple of $5$ i.e $$af+bg+ch+id+ej=5N$$ where $N$ is some positive integer
after that i have no idea how to proccede further
Please help
Thank you
Hm, I guess the maximal cardinality of $S$ is 6. Take any nonzero vector $v$ and (using Gram-Schmidt) construct an orthogonal basis $\mathfrak{B}$ of $V$ from it. Let $S=\mathfrak{B}\cup 0$ be the union of the set of basis vectors (of which there are 5) and the zero vector. Obviously, there can't be a bigger set as each pair of vectors must be orthogonal. This works over all fields which makes me think that this is either a trick question or I have misunderstood it.