Canonic identification $T_{I_n}\mathfrak{gl}(n,\mathbb{R}) \rightarrow \mathfrak{gl}(n,\mathbb{R}) $ I am currently studying Lie Groups and have a questions about this: Let $x_{ij} : M_n(\mathbb{R}) \rightarrow \mathbb{R} \\\ ~~~~~~~A \mapsto A_{ij}$ Then it says let $f$ be _the canonic identification_ $$f: T_{I_n}\mathfrak{gl}(n,\mathbb{R}) \rightarrow \mathfrak{gl}(n,\mathbb{R}) \\\ f(A)_{ij} = A(x_{ij}) ~~~ \forall A \in T_{I_n}\mathfrak{gl}(n,\mathbb{R})$$ I would like to know what _canonic identification_ means in this context. I appreciate any help.

I think what is meant is that for a vector space $V$ (such as $\mathfrak{gl}(n,\mathbb R)$), and a point $x\in V$, there is a natural identification of $T_xV$ with $V$. From your comment I take it that you prefer to view tangent vectors as operators on functions. In this language, the identification maps $v\in V$ to the directional derivative at $x$ in direction $v$, i.e. to the operator $\phi\mapsto \frac{d}{dt}|_{t=0}\phi(x+tv)$, which is an element of $T_xV$.