NUS-MATH530 5.A and Discussion

Chapter 4 discussion with Lachlan 4.2 False.
The union between \{0\} \cup \{p \in \mathcal{P}(\mathbb{F}): deg\ p = m\} is not closed under addition. You can add two m degree polynomials and get something that’s not m degrees:

\begin{equation} (z^{m} + 1) - z^{m} = 1 \end{equation}

4.3 False.
The union between \{0\} \cup \{p \in \mathcal{P}(\mathbb{F}): deg\ p\ even\} is not closed also under addition, for the same reason:

\begin{equation} (z^{m} + z^{m-1} + 1) - (z^{m} + 1) = z^{m-1} \end{equation}

One Chapter 5 Exercise 5.A.5 Suppose T \in \mathcal{L}(V), prove that the intersection of every collection of V that is invariant under T is invariant under T
Let U_1 \dots U_{n} be invariant subspaces under T.
That is:

\begin{equation} T u_{j} \in U_{j} \end{equation}

We desire that:

\begin{align} Tu \in \bigcap U_{j}\ |\ u \in \bigcap U_{j} \end{align}

WLOG, treat u \in \bigcap U_{j} as u \in U_{j}. Now, Tu \in U_{j}. This holds \forall U_{j}. Therefore, Tu \in \forall U_{j}. So Tu \in \bigcap U_{j}.
Hence, the intersection of invariant subspaces are invariant as well.