Let $ V $ be a finite dimensional vector space over a Field $ F $ and let $ T \in End(V) $.
i) If $ S \in End(V) $ is such that $ ST = TS $, show that $ Im(S) $ is a $T$-Invariant Subspace of $ V $
My approach involved showing that $Im(S)$ is a subspace of $V$ such that $T(Im(S))$ is a subspace of $ Im(S)$. Given the property that both $T$ and $S$ are Endomorphisms they map to subspaces of themselves. From this I derive that $T(Im(S))$ is a Subspace of $Im(S)$ which is a Subspace of $V$.
Is this the right approach? There are also two other parts I'm fairly clueless about.
ii) Suppose that $ST = TS$ for all $S \in End(V)$. Show that every 1-dimensional subspace of $V$ is $T$-invariant.
iii) With $T$ as in (ii), show that $T =λI$, for some $λ∈F$.
Could someone point me in the right direction and show me the specific tools needed to answer these question, much appreciated.
Answer-Hint
For i) let $y\in Im(S)$ so $y=S(x)$ for some $x\in V$ and then $T(y)=TS(x)=S(T(x) )\in Im(S)$ hence we proved that $T(Im(S))\subset Im(S)$ which means that $Im(S)$ is $T$-invariant.
For ii) it suffices to take $S$ such that $Im(S)$ is $1$-dimensional subspace of $V$ and we apply i).
For iii) prove that if $T(x)=\lambda_x x$ (this is assumption is deduced from ii)) then $T=\lambda I$ which means that $\lambda_x$ is independent of $x$.