Suppose there is a simple graph, and each vertices are colored white at the beginning. Now you can do the following operation, select a vertex, and flip the color of it and it's adjacent vertices. (White->Black, Black->White)
Now I want to show that I can always flip the color to black for all vertices.
This problem can be also viewed as the following: The 0,1 adjacent matrix $G$ (satisfy $G_{ii}=1$) of a simple graph contains $(1,...,1)$ in the range space in $F_2$. But I don't know how to prove it if $G$ is not invertable.
Either linear algebra or combinatorial proof are welcome.