Basic sanity check: dimension of Lie groups / tangent spaces

Question

A potential typo in an exercise prompted me to question my knowledge of manifolds. So what I need is a sanity check.

Here is what I used to think before I got unsure:

If $M$ is an $n$-manifold then the dimension of the tangent space $T_p M$ at any point $p$ is equal to $n$. As a consequence, since $GL_n$ and its subgroups are manifolds, it must be that if $G$ is an $n$-dimensional matrix group then its Lie algebra $\mathfrak g$ must have dimension $n$ also. (since Lie algebra is defined to be the tangent space at $I$)

Is this correct or not?

Answer 1

Yes, that's correct. ${}{}{}{}{}{}{}{}$

Answer 2

Yep. Sanity checks are often a good idea.