V is a finite dimensional vector space. I am given a linear map $E:V\rightarrow V$ which is a projection so that $E^2=E$. I am also given that $V=Im E\oplus KerE$ (which follows from it being a projection anyway). $E^*$ is the adjoint linear map.I have shown if $E=E^*$ then $(ImE)^\perp=KerE$. I need to show the reverse direction.But I am stuck. I have got as far as showing that $KerE=Ker(E^*)$ so the restrictions of $E $and $E^*$ restricted to the Kernels are equal. It then follows that$ ImE=Im(E^*)$ You can get the orthogonal complements of the images are equal and because we are working in a finite dimensional vector space $V$ we can apply the fact that the orthogonal complement of the orthogonal complement of the subspace gives back the original subspace and so $ImE=Im(E^*)$. I spent a bit of time trying to show the stronger result that the images were equal AND the stuff in both sets was mapped from the same stuff by $E$ and $E^*$ showing $E$ and $E^*$ were the same but I couldn't do this.
By adjoint $E^*$ I mean the map such that $\langle Ev,w \rangle= \langle v,E^*w \rangle \forall v,w \in V$ I forgot to add V is an inner product space, sorry.
