Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by pi-radians

Proof:
Here is a simple view for this problem.
V=R^2 is a 2-dimensional space. If W is a subspace of V, then W is a line passing
through the origin (0,0). T is a rotation clockwise about the origin by pi-radians,
then T maps (x,y) to (-x,-y). Actually, T is a reflection about the origin.
Thus TW=W. Furthermore, (T^k)W=W. T(x,y)=(-x,-y), T^2(x,y)=(x,y).
Suppose f(x) is any polynomial in F[X], then f(T)W=W.
Because for any point (x,y) in W, assume f(x)=a0+a1x+...+anx^n, then
f(T)(x,y)=(a0+a1T+...anT^n)(x,y)=(a0-a1+a2-...)(x,y)=k(x,y) for some k in F. Thus
f(T)(x,y) is in W. So f(T)W=W.
Therefore, W is a F[X]-submodule for this T.