In a finite field $\{0,1,2\}^2$, given a set of vectors $[0\:1],[1\:0],[1\:1],[2\:2]$, we can have the linear combination, $c_1[1\:0]+c_2[0\:1]+c_3[1\:1]+c_4[2\:2] = [s_1\:s_2]\in\{0,1,2\}^2$, where $c_1,c_2,c_3,c_4\in \{0,1,2\}$.
The basis of this set of vectors is $\{[0\:1],[1\:0]\}$. The dimension of this basis is 2, hence this set of vectors span the entire finite field $\{0,1,2\}^2$.
However if the value each constant can take now changes to $c_1\in\{0,1\}, c_2\in\{0,1,2\}, c_3\in\{0,1\},c_4\in\{0,1\}$, is there a more efficient method to determine if these set of vectors still span the finite field $\{0,1,2\}^2$, without rigorously checking if each vector from the finite field $\{0,1,2\}^2$ can be achieved from the linear combination,
$c_1[1\:0]+c_2[0\:1]+c_3[1\:1]+c_4[2\:2] = [s_1\:s_2]\in\{0,1,2\}^2$, where $c_1\in\{0,1\}, c_2\in\{0,1,2\}, c_3\in\{0,1\}, c_4\in\{0,1\}$ ?