Primary Decomposition theory

Dears,

Could you please help me in proving the following:

" Let $\displaystyle T$ be a linear opearator on finite dimensional spsce $\displaystyle V$, let $\displaystyle p = p_1^{{r_1}} \cdots p_k^{{r_k}}$ be a minimal polynomial for $\displaystyle T$, and let $\displaystyle V = {W_1} \oplus \cdots \oplus {W_k}$ be the primary decomposition for $\displaystyle T$, i.e., $\displaystyle W_i$ is the null space of

$\displaystyle {p_i}{(T)^{{r_i}}}$ . Let $\displaystyle W$ be any subspace of $\displaystyle V$ which is invarient under $\displaystyle T$. Prove that

$\displaystyle W = (W \cap {W_1}) \oplus (W \cap {W_2}) \ldots \oplus (W \cap {W_k})$ "

With Best Wishes

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Re: Primary Decomposition theory

The canonical decomposition of a vector space with respect to a linear transformation T can be found in any good linear algebra book. The solution to your problem is really just understanding this decomposition:

Attachment 27899