Let $\phi \in \operatorname{End}(V) , V = \mathbb R^3, S = \{ s_1,s_2,s_3\} $ $$D_S(\phi)=\begin{pmatrix} 1 & 1&2\\0&1&0\\0&0&-1\end{pmatrix}$$ $ W =\langle e_1\rangle, T=\{e_2 +W,e_3+W\}$
How do i get $D_{\{e_1\}}(\phi\big|_W)$ and $D_T(\bar\phi)$ with $\bar\phi: V/W \longrightarrow V/W, \bar\phi(v+W)=\phi(v) +W ?$
I would say $D_{\{e_1\}}(\phi\big|_W)$ = $\begin{pmatrix}1&0&0\\0&0&0\\0&0&0 \end{pmatrix}$ since $\begin{pmatrix} 1\\1\\0\end{pmatrix}$ and $\begin{pmatrix} 2\\0\\-1\end{pmatrix} \notin W$ but im totally unsure.
Note that $D_{\{e_1\}}(\phi_W)$ is the matrix of a transformation on a $1$ dimensional space, so it should be $1 \times 1$. Verify that $D_{\{e_1\}} = 1$.
The second one a bit trickier. However, note that $$ \bar \phi (e_2 + W) = (1) e_2 + (0)e_3 + W $$ as such, the first column of the matrix $D_{T}(\bar \phi)$ should be $(1,0)$.