Proof of invertibility of $I+S$ such that $S^* = -S$ We're given a linear transformation $S:V \rightarrow V$ on a finite dimensional inner product space $V$ such that its adjoint is negative $S$ (i.e. $S^* = -S$). We are able to prove that $(v+Sv,v+Sv)=(v,v) + (Sv,Sv)$ and that the kernel of $(I+S) = \{\mathbf{0}\}$.
Using these facts how can we deduce that the linear transformation $I+S$ is invertible?
Any help would be much appreciated!

If $T \colon V \rightarrow V$ if a linear map on a finite dimensional vector space with trivial kernel then $\dim \ker T + \dim \operatorname{Im} (T) = \dim V$ implies that $\dim \operatorname{Im}(T) = \dim V$ and so $T$ is also onto and invertible.
In your case, if you have shown that $I + S$ has trivial kernel it implies that $I + S$ is invertible.
