Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside in their pile $P_i$ (we also remove this vector from $X$). If at the end of their turn, there exists $m$ vectors in $P_i$ ($m \le n$) that are linearly dependent, then player $i$ loses.
I was wondering if there has been any work on matrix/nim games such as these, and any papers or suggestions would helpful.
Edit: I am only interested in the case where $X$ results in no draws.
A game I am particularly interested is the following. Let $X$ contain only the vectors with everything $0$ except for $2$ positions that have $1$'s ($n \choose 2$ of them). Now play the following game above, and set $m=3$.
Note I am also interested if there is a way to convert, or embed such a game into an impartial game (So Sprague-Grundy applies).
Here is another perspective on the game you are particularly interested in. Suppose there exist 3 linearly dependent vectors in a pile. Then those vectors must be those with non-zero indices $(a_1, a_2)$, $(a_2,a_3)$, and $(a_1,a_3)$. So we can reformulate the game as follows:
There is a graph with $n$ vertices and no edge. On their turn, each player connect two vertices with their color. Player $i$ loses if there exists an $i$-colored triangle at the end of their turn.
If $n \leq 5$, there can be draws. But if $n \geq 6$, we know there must be a triangle of one color, hence a winner.