Proving subspaces are invariant for different statements

Question

I would like to know how to prove the next statements regarding invariant subspaces:

Statement 1:

$f$ and $g$ are endomorphisms from a vector space $V$. If $f$ and $g$ commute, then subspaces $Ker(f)$ and $Im(f)$ are $g$-invariant.

Statement 2:

Based on the previous statement, prove that the general subspaces associated to the eigenvalues of a endomorphism $f$ are $f$-invariant.

Statement 3:

If $U$ and $W$ are invariant subspaces of a $f$ endomorphism, then the subspaces $U\cap W$ and $U + W$ are $f$-invariant.

Thank you very much.

Answer 1

• Statement 1: Let $x\in \ker f$ then $f(g(x))=g(f(x))=g(0)=0$ so $g(x)\in\ker f$. Now let $y\in \rm{Im}(f)$ so $y=f(x)$ and then $g(y)=g(f(x))=f(g(x))\in\rm{Im}(f)$.
• Statement 2: Notice that $f-\lambda \rm{id}$ commutes with $f$ then $\ker(f-\lambda\rm{id}$) is $f$-invariant.
• Statement 3: It's in fact clear but let's prove the second result:

Let $z\in U+W$ so $z=u+w$ where $u\in U$ and $w\in W$ hence $f(z)=f(u)+f(w)\in U+W$ since by hypothesis $f(u)\in U$ because $U$ is $f$-invariant and $f(w)\in W$ because $W$ is $f$-invariant.