In one of the paper I'm reading, the following change of variables is used:
"Knowing $W$, we now choose a change of variables sending $W$ to the last $o_2$ coordinates in $\mathbb{F}_q^m$."
I am trying to work out an example, here are my parameters: $q=4$, $n=4$, $m=2$, $o_2=\text{dim}(W) = 1$. We are working over $\mathbb{F}_4:=\mathbb{F}_2/(x^2+x+1)$ and I denote the root of $x^2+x+1$ by $\alpha$. The vector space $W$ looks like this:
$W = \left\{\begin{pmatrix}1\\ \alpha+1\end{pmatrix}\right\}$. I am thus looking for a change of variables $\phi:\mathbb{F}_4^2\to\mathbb{F}_4^2$ that sends the vector $w=\begin{pmatrix}1\\ \alpha+1\end{pmatrix}$ to $e_2$ I assume? So does the map $e_1\to e_1$, $w\to e_2$ work?