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I want to prove the following,

Let $T$ be a linear transformation on a finite dimensional vector space $V$ with dim$V \neq 0$. Then for some integer $m \geq 1$, there exist $T$-invariant subspaces $U_1,...,U_m$ which are indecomposable with respect to $T$ , such that $V = U_1 \oplus \ ... \oplus \ U_m$.

$\oplus$ denotes a direct sum.

Some background:

Let $U$ be a subspace of a finite dimensional vector space $V$. Let $T : V \rightarrow V$ be a linear transformation.$ \ U$ is said to be $T$-invariant if $T(U) \subseteq U$, where $T(U)$ denotes the image of the subspace $U$ under $T$.

Let $U$ be a subspace of a finite dimensional vector space $V$. Let $T : V \rightarrow V$ be a linear transformation. A subspace $U$ of $V$ is said to be decomposable with respect to $T$ if there exist proper subspaces $U_1,U_2$ of $U$, invariant under $T$ such that $U = U_1 \oplus U_2$.

I have been given a hint, to consider the proof that any integer can be factorised into primes, and to imitate this. I am referring to the proof here: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

The reason why I want to prove this lemma is that it is useful for proving that any linear mapping can be represented as a Jordan matrix. It is a chain of one of many lemmas to arrive at the result. It's also a pretty nice result that draws parallels with the factorisation of an integer into prime numbers.

My attempt:

Let $V$ be a non-zero, finite dimensional vector space. Let $T : V \rightarrow V$ be a linear transformation. We can write $$V = V \oplus \{\textbf{0}\}$$ where $\textbf{0}$ denotes the zero element of $V$. It is clear that $\{\textbf{0}\}$ is indecomposable. If $V$ is indecomposable, then we are done, so we suppose $V$ is decomposable, so we can write $$V = U_1 \oplus U_2$$ where $U_1$ and $U_2$ are proper subspaces of $V$, and $U_1$ and $U_2$ are $T$-invariant. I could keep going with this line of reasoning, saying if $U_1$ and $U_2$ are indecomposable, we are done. So suppose at least one of $U_1$ and $U_2$ are decomposable.

I know that this process can't go on indefinitely, since $V$ is finite dimensional, we have $$\sum_{i = 1}^{m} \text{dim}(U_i) = \text{dim}(V)$$ This suggests to me that I may need to use induction, but I am not sure how to implement it here.

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If $V=U_1\oplus U_2$ is a nontrivial decomposition then both $U_1$ and $U_2$ have smaller $\dim$ than $V$.

With this in mind, do you see what the induction hypothesis should be?

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