Let $F_p = Z/pZ$ be the field with p elements for some prime p, and consider the vector space $V = F^3_p$ over $F_p$. Find an ordered basis for V containing the element $(1; 1; 1)$.
How does one go about finding a basis for an arbitrary modulo vector space?
An easier set of vectors to work with is $(1,1,1),(0,1,0),(0,0,1)$.
In order to show that the vectors $v_1,v_2,v_3 \in \Bbb F^3$ are linearly independent, it suffices to show that the matrix $(v_1 ,v_2,v_3)$ is nonsingular. Try taking the determinant.