If $2$ elliptic curves over a field with characteristic different from $2$ and $3$ are isomorphic via an invertible linear map $\mathbb P^2\to\mathbb P^2$, then how does this map look like ? (the task is to prove that their $j$-invariant is equal, this follows much easier from a theorem I think, but here I have to use the linear map)
Say elliptic curves are $E_1:y^2=x^3+a_1x+b_1,\quad E_2:y^2=x^3+a_2x+b_2$ and then the linear map is just a $3\times 3$ matrix
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}c&d&e\\f&g&h\\i&j&k\end{pmatrix}\begin{pmatrix}x'\\y'\\z'\end{pmatrix}$
Now, my question: Why is $x=x',y=y',z=z'$ ?