I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my proof (i.e. U is T-invariant => $P_UT=TP_U$).
I'd appreciate suggestions to:
- Reduce unnecessary verbiage and make my proofs more concise and readable
- Remove any possible logical flaws
Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. Prove that $U$ and $U^\perp$ are invariant under $T$ if and only if $P_UT = TP_U$.
Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. By Theorem 6.32, $V = U \oplus U^\perp$.
Suppose $U$ and $U^\perp$ are invariant under $T$. Let $v \in V$. By direct sum decomposition, we can write $v = u + w$ where $u \in U$ and $w \in U^\perp$. Then, $P_UTv=P_UT(u+w)=P_UTu+P_UTw$. Since $Tu \in U$, we have $P_UTu=Tu$. Since $Tw \in U^\perp$, we have $P_UTw=0$. Therefore, $P_UTu+P_UTw = Tu+P_UTw=Tu$. Since $u \in U$, $u=P_Uu$. Therefore, $Tu=TP_Uu$. Since $w \in U^\perp$, $P_Uw = 0$ and so $T0=TP_Uw=0$. So $P_UTv=Tu=Tu+0=TP_Uu+TP_Uw= TP_U(u+w)=TP_Uv$. Therefore, $PT_Uv=TP_Uv$ as we needed to show.
Second part:
Now suppose $P_UT=TP_U$. Let $u \in U$ and $w \in U^\perp$. By definition of projection, $P_Uu=u$ and $P_Uw=0$. We have $P_UTu=TP_Uu=Tu$. By definition of projection $P_UTu \in U$. $Tu \in U$ and therefore $U$ is T-invariant. By definition of projection, $P_Uw=0$. We have $TP_Uw=T0=0=P_UTw$. Since $P_UTw=0$, we know that $Tw \in U^\perp$ and therefore $U^\perp$ is also T-invariant.
Thanks again!
Not bad at all. However, you only proved half the statement! You need to prove that if $P_UT=TP_U$, then $U$ and $U^\perp$ are $T$-invariant.
Remember that you are writing for an audience (that's us!), and that your job is to communicate with us as clearly as possible while being succinct. The long string of equalities is a bit much in the second to last sentence of your proof. You can significantly cut down on the wordage and do most of the grunt work earlier in your proof. While everything you have written down is right, there is no clear flow to your proof. You calculate a bunch of quantities and then show how they give the result at the end. I would structure your proof by saying something like, "first, we find $P_UTv$", and then do the related calculations to conclude $P_UTv=Tu$ (easy!). Then tell us, "now, we will find $TP_Uv$" and conclude after some more calculations that $TP_Uv = Tu$ (also really easy!) You've shown that $P_UTv$ and $TP_Uv$ are equal to the same thing, so you can point that out to us, and then you're done with that direction of the proof!