Let $V$ be an m-dimensional vector subspace of $\mathbb{R}^{k}$, let $p \in \mathbb{R}^k$, and let $M=p+V$. Similarly, let $W$ be an n-demensional vector subspace of $\mathbb{R}^{l}$, let $q \in \mathbb{R}^{l}$, and let $N=q+W$. Let $T:V \rightarrow W$ be a linear map. Define $\bar{T}:M \rightarrow N$ satisfying $$ \bar{T}(p+v)=q+T(v) $$ That M and N are manifolds with $TM_x=V$ and $TN_x=W$ can easily be showed. Now we prove that
a) $\bar{T}$ is the smooth function on M in the sense that for each $x \in M$, there exists an open set $U \subset \mathbb{R}^{k}$ containing $x$ and a function $F: U \rightarrow \mathbb{R}^{l}$ such that $F \in C^{\infty}(U)$ and $F_{|U \cap M}=\bar{T}_{|U \cap M}$
b) Find the derivatives of $\bar{T}$ at $x \in M$. This means that we have to look for $dF_{x}(h), h \in TM_x=V$
I struggled to find the smooth extension $F$ of $\bar{T}$. With $x=p+l$ , I've set $$ F(a)= \begin{cases} q+T(a-p) \hspace{4mm},a \in B(x,\epsilon) \cap M= p+V \cap B(l,\epsilon) \\ g(a) \hspace{20.8mm} ,a \in B(x,\epsilon) \cap M^c= p+V^{c} \cap B(l,\epsilon) \end{cases} $$ and then tried to find an $\epsilon>0$ and appropriate $g:p+V^{c} \cap B(l,\epsilon) \rightarrow \mathbb{R}^{l}$ such that $F \in C^{\infty}(B(x,\epsilon))$, but I have no idea what to do next.
Any help would be appreciated!!
For a) you could first extend your linear map $T:V\to W$ to a linear map $T^{e}:\mathbb R^k\to \mathbb R^l$ (which then will be smooth) and define $F(a)=q+T^{e}(a-p)$.