I have the following problem:
Let $K$ be a field and let $V$ be a finite dimension $K$-vector space. Let $T \colon V \longrightarrow V$ be a linear transformation with minimal polynomial of degree $m$. Show that $V$ has a $T$-invariant subspace of dimension $m$.
I wanted to use that $V$ is isomorphic to $K[t]$-cyclic modules of the form $$V \cong \frac{K[t]}{\langle f_1\rangle} \oplus \frac{K[t]}{\langle f_2\rangle} \oplus \cdots \oplus \frac{K[t]}{\langle m_T\rangle}$$ where $f_i$ divides $f_{i + 1}$, $f_i$ is the minimal polynomial of the corresponding restriction of $T$ and $m_T$ is the minimal polynomial of $T$. But I got stuck.
Is my way correct? Or am I getting more complicated than neccesary? Thanks in advanced!