= Solution
The <primary decomposition theorem for linear maps> starts with a <linear map> $T:V\to V$ on a <finite-dimensional vector space> over a <field> $F$. Factor its <minimal polynomial> as
$$
m_T(x)=\prod_{j=1}^r p_j(x)^{a_j},
$$
where the $p_j$ are distinct <monic polynomials> that are <irreducible polynomials> and $a_j\geq1$. Define $V_j=\ker p_j(T)^{a_j}$. Then
$$
\boxed{V=V_1\oplus\cdots\oplus V_r,\qquad T(V_j)\subseteq V_j,\qquad m_{T|_{V_j}}=p_j^{a_j}}.
$$
Thus the <direct sum> separates the <irreducible polynomial> factors into uniquely determined <invariant subspaces>. The statement works even when the <minimal polynomial> does not split into <linear polynomials> over $F$.
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