# Writing a vector space as a direct product of T-irreducible subspaces

Let $$V$$ be a finite dimensional vector space over a field $$K$$ of characteristic zero, and let $$T \in End(V)$$. A subspace $$W \subseteq V$$ is $$T$$-invariant if $$T(W) \subseteq W$$; and $$W$$ is $$T$$-irreducible if it is T-invariant and the only T-invariant subspaces of $$W$$ are $$0$$ and $$W$$.

(a) Suppose that $$T \in End(V)$$ has order $$n$$ in $$End(V)$$ under composition, and that $$W \subseteq V$$ is a T-invariant subspace, Show that $$W$$ has a T-invariant complement $$W'$$.

(b). Under the hypotheses of $$(a)$$, show that $$V$$ can be written as the direct product of T-irreducible subspaces.

(c). What if $$T$$ does not have finite order in $$End(V)$$?

(d). What if $$K$$ does not have characteristic zero?

I was able to prove parts (a) and (b). However, I am not so sure about parts (c) and (d). Since my proof of parts (a) and (b) relied on the fact that $$T$$ had finite order and $$K$$ had characteristic zero, I assume that the results no longer hold. This all seems similar to to Maschke's theorem, but I am not familiar with representation theory.

The matrix $$T:=\pmatrix{1&1\\0&1}$$ provides a counterexample for both c) and d).
• Well, set $W:={\rm span}\pmatrix{1\\0}$. Show that it doesn't have a $T$-invariant complement. Nov 28, 2021 at 9:53