At the moment I am studying Lie theory while self-studying differentiable manifold (by Lee's book), and I got stuck at understanding what tangent space and vectors really are especially while going through Lie theory.
First thing I got confused with is this:
I know two ways to understand tangent spaces, one via equivalence classes of curves (namely, if two curve has identical derivative after projecting the curve onto Euclidean space via coordinate charts, then we view the equivalent classes of curves as tangent spaces), and other via a set of derivations over the smooth manifold. In our lie theory course, we defined $\alpha'(0) = \frac{d\alpha}{dt}|_{t = 0}$, and I used to believe that this definition just means that we are taking equivalent classes over the value of derivative at (t = 0) (which is independent of the choice of the coordinate chart), and thus $\alpha'(0)$ represents a curve passing through particular point on the manifold.
However, when we tried to actually compute out Lie algebra, I saw that lie algebra of $GL(n, \mathbb{R})$ is $M(n, \mathbb{R})$, which is a tangent space at identity. However, the set of tangent vectors on identity, $M(n, \mathbb{R})$, is neither curve nor derivation. Plus, tangent space is not an element of lie group (smooth manifold) G, but we can take one parameter subgroup $\phi(t) = e^{tX}$ where $X$ is an element of tangent space...??? Since one parameter subgroup takes in real number and maps to smooth manifold, and in this case since one parameter subgroup has $X$ in its function which is an element of tangent space (i.e. lie algebra) not manifold, how could I understand this situation? In those cases, how could I understand what object precisely tangent space is and for this specific case, how could I make sense out of it?
Thank you very much!!
You are thinking about tangent spaces on a too high level for this example. Since $GL(n,\mathbb R)$ is an open subset in the vector space $M_n(\mathbb R)$ of $n\times n$-matrices any tangent space can be identified with the ambient vector space. Explictily, given $X\in M_n(\mathbb R)$ and $A\in GL(n,\mathbb R)$, this corresponds to the equivalence class of the curve $c(t)=A+tX$ (which lies in $GL(n,\mathbb R)$ for sufficently small $|t|$. In the picture of deriviations $X$ corresponds to $f\mapsto Df(A)(X)$, the directional derivative in direction $X$ (which again makes sense since $f$ is defined on an open subset). In fact, for matrix groups, you can always think about tangent spaces in the sense of smooth submanifolds of $\mathbb R^N$, so the tangent space will always be a linear subspace of the ambient vector space. (Described for example by all possible values of $c'(0)$ for appropriate curves.)