Suppose $V$ is a 2 dimensional real vector space. Then, there exists an linear isomorphism $x:\,V\longrightarrow\mathbb R^2$. For any pairs of bases $\mathcal V=(v_1,v_2)$ of $V$ and $\mathcal B=(e_1,e_2)$ of $\mathbb R^2$, we define \begin{align} \big[x\big]_{\mathcal V,\mathcal B} = \begin{bmatrix} \big[x(v_1)\big]_{\mathcal B} & \big[x(v_2)\big]_{\mathcal B} \end{bmatrix} \end{align} to be the matrix representation of $x$ respect to the bases $\mathcal V,\,\mathcal B$. In this topic, we will consider $\mathcal B$ to be the standard basis of $\mathbb R^2 $.
Moreover, each isomorphism is also a diffeomorphism between $V$ and $\mathbb R^2$. From there, if $T$ is a tangent vector at $p\in V$, then there exists a smooth curve $\alpha:\,I\longrightarrow V$ such that $p=\alpha(t_0)$ and \begin{align} \\ T\,&=\,(x\circ\alpha)_1'(t_0)\frac{\partial}{\partial u_1}\bigg|_p + (x\circ\alpha)_2'(t_0)\frac{\partial}{\partial u_2}\bigg|_p \newline T(f)\,&=\,(x\circ\alpha)_1'(t_0)\frac{\partial\big(f\circ x^{-1} \big) }{\partial u_1}\big(x(p)\big) + (x\circ\alpha)_2'(t_0)\frac{\partial\big(f\circ x^{-1} \big) }{\partial u_2}\big(x(p)\big)\tag1 \end{align} for all $f\in C^\infty(p) $.
However, since $x:\, V\longrightarrow\mathbb R^2$ is linear, we can write \begin{align} x(p) = x\big(p_1v_1+p_2v_2 \big)=p_1x(v_1)+p_2x(v_2).\tag2 \end{align} This implies \begin{align} \big(x\circ\alpha\big)(t)&=x\big(\alpha(t)\big)=\alpha_1(t)e_1+\alpha_2(t)e_2 \newline (x\circ\alpha)_1'(t_0) &= \alpha_1'(t_0) \newline (x\circ\alpha)_2'(t_0) &= \alpha_2'(t_0).\tag3 \end{align} Now, I wonder if this fact is true \begin{align} \begin{pmatrix} \left[\displaystyle\frac{\partial}{\partial u_1}\bigg|_p \right]_{\mathcal C} & \left[\displaystyle\frac{\partial}{\partial u_2}\bigg|_p \right]_{\mathcal C} \end{pmatrix}= \big[x\big]_{\mathcal V,\mathcal B}\tag4 \end{align} where $\mathcal C$ is some basis of $TV_p$ ? And if we consider the differential at $p$ \begin{align} dx_p:\ TV_p & \longrightarrow \mathbb R^2 \newline T & \longmapsto dx_p(T)\tag5 \end{align} since $dx_p$ is a linear isomorphism, then does the its representation matrix is the same to $\big[x\big]_{\mathcal V,\mathcal B} $ for some pair of bases ?
Thanks.