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Suppose $M=GL_2(\mathbb{R})$ is a smooth manifold. Define left-invariant vector fields $X$, $Y$ on $M$ and left-invariant 1-form $\omega$ on $M$, where

$X_1= \begin{pmatrix} 0 & 1\\ 0 & 0\\ \end{pmatrix}$, $Y_1=\begin{pmatrix} 0 & 0\\ -1 & 0\\ \end{pmatrix}$, and $\omega_1\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+b-d$.

Here, $1\in G$ is the identity of the Lie group. The question requires to calculate $d\omega(X,Y)$ as a function on $M$.

For so far I can exam, there is $d\omega(X,Y)=X(\omega(Y)) - Y(\omega(X)) - \omega([X,Y])$, which could simplify the computation. However, I did not find a good way to compute the vector fields and 1-form at random point of $M$. Could anybody provide a solution to this question as well as demonstrating how to compute the tangent space and cotangent space on random point of $M$?

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$\mathrm{GL}_2 (\mathbb{R})$ is an open subset of $\textrm{Mat}_{2 \times 2} (\mathbb{R}) \cong \mathbb{R}^4$, so the tangent space is a 4-dimensional vector space and can be canonically identified with the space of $2 \times 2$ matrices. Finding the invariant vector fields is a bit tricky, but you can see the calculations in Example 2.4.6 of the notes here. –  Zhen Lin Sep 16 '12 at 5:20

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