$C$ is an $F_3-[5, 2, 3]$ code with generator matrix
$G = \begin{bmatrix}1 & 2 & 0 & 2 & 2\\2 & 0 & 1 & 0 & 2\end{bmatrix}$
Which of the following matrices is the generator matrix of a ternary code that is equivalent to $C$?
$G_a = \begin{bmatrix}1 & 0 & 2 & 0 & 1\\0 & 1 & 2 & 1 & 2\end{bmatrix}$
$G_b = \begin{bmatrix}1 & 1 & 0 & 2 & 2\\0 & 1 & 2 & 2 & 1\end{bmatrix}$
From what I understood, if by applying some monomial transformation to a generator matrix $G'$ you could obtain $G$, then the codes generated by those two matrices are equivalent. In other words, if I could perform permutations of the columns, and multiply the columns by non-zero scalar, of $G_a$ or $G_b$ to transform either into $G$, then the corresponding codes are equivalent.
I cannot seem to find any such transformation from either $G_a$ or $G_b$ to $G$, but one of their codes definitely is equivalent to that of $G$.
Using permutations of columns and multiplying columns by non-zeros scalars, I can get $G_a$ to
$\begin{bmatrix}1 & 2 & 0 & 0 & 2\\2 & 0 & 1 & 1 & 2\end{bmatrix}$
Can I then subtract col3 from col4 to get $col4 = \begin{bmatrix}0\\0\end{bmatrix}$ then add to it col 2 which would then be equal to $G$? (In other words - can I replace col4 with col2?)