How do I check whether a set is a subspace over the modular numbers when said set is defined using integers?

Question

Let $F$ either be $\mathbb R$ or $\mathbb Z/2\mathbb Z$.

For each of the following subsets of $F^3$, determine whether it is a subspace. Be sure to explain your answer. $U_1=\{(x_1,x_2,x_3)\in F^3: x_1+2x_2+3x_3=0\}$

I completed the $\mathbb R$ case, but I'm confused on how to approach the $\mathbb Z/2\mathbb Z$ case. In particular, how is $x_1,x_2,x_3$ a modular number when $U_1$ is define with integers 1,2, and 3?

I'm stuck here: Choose an arbitrary $(x_1,x_2,x_3) \in U_1$. We know $([0],[0],[0]) \in U_1$, since $[0] + 2[0]+3[0]=0$ ... wait but how is that possible ...? they're different types of numbers...?