Tangent Space of Torus

Question

Let T be a torus (namely, topology homeomorphic to quotient topology $\mathbb{R}/\mathbb{Z}$). Intuitively it makes sense to say that tangent space of torus at identity (i.e. lie algebra of 1D torus) is $\mathbb{R}$ since we can geometrically think of tangent space as straight line passing through the torus at identity. However, how could we formally prove this fact? It seems like if I formally want to prove it, I need to consider all derivation maps (or curve passing through origin) then somehow need to show that the set of derivations correspond to the reals, but I am not very sure how the argument works here. Also, how is tangent space of k-dimensional torus like? Could anyone help me to figure out this? Thank you a lot for all of your support in advance!