Invariant Subspace proof

Oct 2019
I am having some trouble with part (ii) of the following question, part (i) is fine;

Let V be a finite dimensional vector space and let f, g : V -> V be linear maps such that f[FONT=MathJax_Main]∘g = id_V.
(i) g[/FONT]
[FONT=MathJax_Main]∘f = id_V
(ii) a subspace of V is f-invariant if and only if it is g-invariant

I've tried the following:
Let W be a subspace of V and let x be in W.
(=>): f(x) is in W since f-invariant
x = g(y) for some unique y in V since g is bijective
f(x) = f(g(y)) = y
so y is in W
therefore g(y) and y are in W

My issue with this attempt is that it's showing that g(y) in W implies y is in W. The definition of invariant means one has to show the implication is the other way around.
All of my attempts have had a similar issue[/FONT]
Jun 2013
first prove that $f(W)=W$. It should be easy after that
  • Like
Reactions: topsquark