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.

Prove:

(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]