The question is given below:
But I do not know how to solve it, if my understanding correct, I see that they want us to find the matrix $A$ in the following homogeneous system $AX = 0$, am I correct? if so, then I still do not know how to find the matrix, could anyone help me in this please? If not, could anyone tell me how to solve this problem please?
Thanks!
Guide:
The nullity is $3$, hence the rank has to be $5-3=2$.
Hence $A \in F^{2 \times 5}$. Let me write $A = \begin{bmatrix} a_1^T \\ a_2^T\end{bmatrix}$ and let the $3$ independent vectors in the solution space be put in a matrix $X$.
$$AX=0$$
means that
$$X^TA^T=0$$
Try to solve the system in $GF(13)$,
$$\left[\begin{array}{ccccc|c} 2 & 1 & 9 & 7 & 4 & 0 \\ 8 & 3 & 10 & 5 & 12 & 0 \\ 7 & 6 & 2 & 11 & 7 & 0\end{array}\right]$$
The solution would help you to reconstruct the matrix $A$.