Faithfull representation of $\mathfrak h \rtimes V$. Let $V$ be a finite dimensional vector space and $\mathfrak h \subset \mathfrak {gl} (V)$ a Lie Algebra. Consider the semidirect product $\mathfrak h \rtimes V$
$$[(H_1,v_1),(H_2,v_2)] = ([H_1,H_2], H_1 v_2 - H_2 v_1). $$
I'm stuck in this question

Question: Let $\mathfrak{h} \subset \mathfrak {gl} (V)$ a Lie Algebra of linear transformations and consider the semidirect product $ \mathfrak g = \mathfrak h \rtimes V $. Find a faithful representation of $\mathfrak h \rtimes V$.

Does anyone have any ideas?
 A: An explicit representation is given by $\rho:\mathfrak{g}\to \mathfrak{gl}(\mathbb R\times V)$,
$$\rho(H,v)=\begin{pmatrix}0 &0_{1\times n} \\ 0_{n\times 1} & H\end{pmatrix}+\begin{pmatrix}0 &0_{1\times n} \\ v & 0_{n\times n}\end{pmatrix}.$$
(here I think on $V$ as $\mathbb{R}^n$, and use matrices).
The verification is straightforward.
More important then this is how to get there. Suppose $H$ is a Lie subgroup of $GL(V)$ that integrates $\mathfrak{h}$. If you think on the natural semidirect product of the groups $H$ and $V$,  $H\rtimes V$, then $H\rtimes V$ acts on $V$ by combining multiplication and translation:
$$(h,v)x=hx+v.$$
You can even define the semidirect product through this action. This action, however, is not linear. You can fix this situation by considering a slightly extended action on $V\times \mathbb{R}$:
$$ (h,v)(x,\alpha)=hx+\alpha v.$$
(note that the last action imitates the first one when you restrict to $\{1\}\times V$.) The last action (magically) happens to be linear. Taking $V=\mathbb{R}^n$, its explicit representation is
$$\tilde\rho(h,v)=\begin{pmatrix}1 &0_{1\times n} \\ v & 0_{n\times n}\end{pmatrix}\begin{pmatrix}1 &0_{1\times n} \\ 0_{n\times 1} & h\end{pmatrix}.$$
You just need to differentiate $\tilde \rho$.
