Let M be a differentiable manifold of dimension n. Consider an atlas \mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}} along with coordinates x_\alpha^1,\cdots,x_\alpha^n in U_\alpha. For x=(x_\alpha^1(x),\cdots,x_\alpha^n(x))\in U_\alpha, a tangent vector is given by
v=\sum_{j=1}^nv_\alpha^j\frac{\partial}{\partial x_\alpha^j}.
If x\in U_\alpha\cap U_\beta, then v is also written as
v=\sum_{j=1}^nv_\beta^j\frac{\partial}{\partial x_\beta^j}.
Here, the change of coordinates is given by
v_\beta^j=vx_\beta^j=\sum_{k=1}^nv_\alpha^k\frac{\partial x_\beta^j}{\partial x_\alpha^k}.
For x\in U_\alpha\cap U_\beta and f=(f^1,\cdots,f^n)\in\mathbb{R}^n, define
\begin{align*}
h_{\alpha\beta}(x)(f)&=\left(\sum_{k=1}^n\frac{\partial x_\beta^1}{\partial x_\alpha^k}f^k,\cdots,(\sum_{k=1}^n\frac{\partial x_\beta^n}{\partial x_\alpha^k}f^k\right)\\
&=\begin{pmatrix}
\frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\
\vdots & \ddots & \vdots\\
\frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n}
\end{pmatrix}\begin{pmatrix}
f^1\\
\vdots\\
f^n
\end{pmatrix}
\end{align*}
Hence h_{\alpha\beta}:U_\alpha\cap U_\beta\longrightarrow\mathrm{Aut(\mathbb{R}^n)}. The resulting bundle over M with fibre F=\mathbb{R}^n is called the tangent bundle of M and is denoted by TM. Note that TM is the set of all tangent vectors of M i.e. TM=\bigcup_{x\in M}T_xM. For each x\in U_\alpha, the fibre \pi^{-1}(x) of x\in M is T_xM\cong\{x\}\times\mathbb{R}^n. The local trivialization map h_\alpha:\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times\mathbb{R}^n is given by
h_\alpha(v)=(x,(v_\alpha^1,\cdots,v_\alpha^n)),\ v\in T_xU_\alpha(=T_xM),\ x\in U_\alpha.
A fibre bundle (E,M,F,\pi) is called a vector bundle over M if each fibre F_x of x\in M is a vector space. So, a tangent bundle is a vector bundle. The tangent bundle TM is a differentiable manifold of dimension 2n with local coordinates in \pi^{-1}(U_\alpha) being (x_\alpha^1,\cdots,x_\alpha^n,v_\alpha^1,\cdots,v_\alpha^n). The Jacobian is given by
\begin{align*}
J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n)&=\frac{\partial(x_\beta^1,\cdots,x_\beta^n)}{\partial(x_\alpha^1,\cdots,x_\alpha^n)}\\
&=\begin{pmatrix}
\frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\
\vdots & \ddots & \vdots\\
\frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n}
\end{pmatrix}:U_\alpha\cap U_\beta\longrightarrow\mathrm{GL}(n,\mathbb{R}).
\end{align*}
Let g_{\alpha\beta}=J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n) Then g_{\alpha\beta} satisfies
\begin{align*}
g_{\alpha\alpha}(x)&=I_n;\\
g_{\beta\alpha}(x)&=g_{\alpha\beta}^{-1}(x),\ x\in U_\alpha\cap U_\beta;\\
g_{\alpha\beta}(x)g_{\beta\gamma}(x)g_{\gamma\alpha}(x)&=I_n,\ x\in U_\alpha\cap U_\beta\cap U_\gamma.
\end{align*}
For x\in U_\alpha\cap U_\beta and f\in\mathbb{R}^n, h_{\alpha\beta}(x)(f)=g_{\alpha\beta}\cdot f. So, \mathrm{GL}(n,\mathbb{R}) acts on the fibre \mathbb{R}^n. The map g_{\alpha\beta} itself is often called a transition map.
If the transition map h_\alpha\beta is the group action of a Lie group G on the fibre F, the fibre bundle (E,M,F,\pi) is called a G-bundle and the Lie group G is called a structure group. The tangent bundle TM is also a G-bundle with structure group \mathrm{GL}(n,\mathbb{R}).