We have T = Id. That T(T(v)) = v for all vin V.
U={
v
V: T(
v)=
v} and W={
v
V: T(
v)= -
v}.
We will show either U orW is a nonzero subspace.
Suppose U = 0 then T(v)
v for all non zero vectors v in V.That is v-T(v) is nonzero vector
Let w = v-T(v) ,
Then T(w) = T(v) - T(T(v))= T(v)- v = - (v - T(v) ) = -w.
That is T(w) = - w which implies w
W and hence W is nonzero.
So either U is nonzero or W is a nonzero subspace of V.
Suppose v
U
W. Then T(v) =v and T(v) = -v. That is v = -v and hence v = 0.
UW ={0}.
Let v
V. Let w = v +T(v). Then T(w) = T(v) + T(T(v)) = T(v) + v = w
That is w
U. Then w/2
U
Let w' =v - T(v). Then T(w') = -w'. Hence w'
W. w'/2
W
Now v = w + w'
U+W.
So V is contained in U + W.Since U and W are subspaces of V, U+W is contained in V
Hence V =U+W.