Why does $T_I GL(V)=End(V)$ hold? (For $I$ the identity matrix and $V$ any fin. dim. vector space over the reals) My book only states that "since $GL(V)$ is an open subset of the linear space $End(V)$ its tangent space may be identified with $End(V)$." But I do not get it
Any ideas why this holds?
Draw a picture of an open set in $\mathbf{R}^2$. Select a point in its interior. What is the tangent space of the open set at that point? It should be all of $\mathbf{R}^2$ right?
Think about it this way: $\det I = 1$ and so if we move a little bit in any direction, say $\varepsilon A$ then $\det(I + \varepsilon A) \approx 1$ still. So $I + \varepsilon A \in \mathrm{GL}(V)$. But this works for any direction. So the curve $f(t) = I + tA$ is, for small $t$, contained in $\mathrm{GL}(V)$. Therefore $f'(0) = A$ is a vector in the tangent space at $I$.