I know that there are two commonly used definitions of the tangent space at a point of a smooth manifold $M$: one using paths through that point, and the other using the idea of linear maps $C^\infty(M)\to\mathbb{R}$ that satisfy the Leibnitz rule for product differentiation. In the Lie Groups course that I'm currently studying, we use the latter definition, so $T_aM$ is the vector space of all tangents vectors at $a$ and has a basis $$\left\{\left(\frac{\partial}{\partial x_i}\right)_a\right\}$$ where we are using coordinates given by $\varphi$, and $$\left(\frac{\partial}{\partial x_i}\right)_a (f) = \frac{\partial f}{\partial x_i}\bigg|_{\varphi(a)}.$$
Most of the answers on this site (all the ones that I can find, at least) talk about how to find the tangent space explicitly using the path definition, but I'm interested in doing it with this definition. I know that they are probably very similar (if not the same, up to notation), but I'm struggling to understand it, and would appreciate a concrete example.
Here is the example I'm trying to work through, with the aim of finding the Lie algebra of $G$ coming from the tangent space at the identity:
Let $G$ be the $3$-dimensional Lie Group of matrices of the following form: $$G = \left\{\left(\begin{array}{cc}A&a\\0&1\end{array}\right) : A\in \mathrm{O}(2),a\in\mathbb{R}^2\right\}.$$ We know that $A\in\mathrm{O}(2)$ can always be written as exactly one of $$\left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right)\quad\text{or}\quad\left(\begin{array}{cc}\cos\theta&\sin\theta\\\sin\theta&-\cos\theta\end{array}\right)$$ for $\theta\in(0,2\pi]$, and so $\psi\colon A\mapsto\theta\det A$ is a coordinate chart on $\mathrm{O}(2)$, giving us the coordinate chart $\varphi\colon B\mapsto(\theta\det A,a_1,a_2)$, where $B\in G$.
So the basis of $T_eG$ acts on $C^\infty(G)$ by $$\left(\frac{\partial}{\partial x_i}\right)_e(f) = \frac{\partial f}{\partial x_i}\bigg|_{\varphi(e)} = \frac{\partial f}{\partial x_i}\bigg|_{(2\pi,0,0)}$$ where $f\in C^\infty(M)$.
Questions:
- Is the above right?
- How can I continue from here (after making any corrections from question 1.).