I'm pretty much stuck on the following question (taken from the book Lie groups and introduction to linear groups by Rossman W.):
I've found some clues, but I think I lack proper understanding of what needs to be done here.
If I recall correctly, the relation between lie groups homomorphism $T$ and its lie algebra homomorphism $\tau$ is $\tau(X) = \frac{d}{dt} T(e^{tX})|_{t=0}$ when in this case $ X \in gl(n,\mathbb{R})$. But then, some factor involving the exponent function should appear. On the other hand, the derivative of $f(x)$ is $$\displaystyle \sum_{i} \xi_i ' \frac{\partial f}{\partial\xi_i}$$, so it should be some kind of chain rule?
In (b) I have similar problem - it seems like I need to "go down" to $so(3)$ and use its basis matrices, but I'm not sure how to justify it.
Any assistance will be highly appreciated.