Let $p\in M$ with $M$ being the group of invertible upper triangular matrices. Determine $T_pM$.
My idea - I know that $M$ is a submanifold of the vector space $U_n$ (upper triangular matrices). Moreover, it is an open subset of it. By a theorem, for every $p\in U_n$, $T_pU_n=U_n$. Now I vaguely remember a theorem that says that if $U\subset M$ is open then for every $x\in U$, $T_xM=T_xU$ (but I wasn't able to prove it). So my feeling is that for our $M$, $T_pM=U_n$.
Is my idea correct? If so, can anyone provide me a proof of the theorem I stated?
The definition of tangent space Im working with is $T_xM=\{[\gamma]_x:\gamma:(-\varepsilon,\varepsilon)\to M, \gamma(0)=x\}$ with $\gamma\sim\delta$ if for every chart $f$ that contains $x$, $(f\circ\gamma)'(0)=(f\circ\delta)'(0)$.
For embedded submanifold, a chart $\phi$ and $y=\phi(x)$, $T_xM=ImD\phi_y$.
Tangent spaces are essentially derivations, so they are local. From your geometric definition of tangent spaces, we can define an isomorphism between $T_x U$ and $T_x M$, by sending $[\gamma]$ to $[\gamma]$. It has an inverse, which is sending $[\gamma]$ to $[\gamma|_U]$. You can show they are both well-defined.