I'm posed with the following problem. Given a vector space $\,V\,$ over a field (whose characteristic isn't $\,2$), we have a linear transformation from $\,V\,$ to itself.
We have subspaces
$V_+=\{v\;:\; Tv=v\}\,\,,\,\, V_-=\{v\;:\; Tv=-v\}$
I want to show that $\,V\,$ is the internal direct sum of these two subspaces. I have shown that they are disjoint, but can't seem to write an arbitrary element in V as the sum of respective elements of the subspaces...
Edit:forgot something important!
$\, T^2=I\,$
Sorry!