# Prove that the minimal polynomial of $T_W$ divides the minimal polynomial of T.

Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a T-invariant subspace of V. Prove that the minimal polynomial of $$T_W$$ divides the minimal polynomial of T.

My try: Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a T-invariant subspace of V. let p(t) be the minimal polynomial of T. For $$w \in W$$, we have $$p(T_w)(w)=P(T)(w)=0$$ by definition of minimal polynomial. Hence proved.

Quotient space: Let V be any vector space and $$W \subset V$$ and buspace. For any $$v in V$$, let $$v+W$$ and denote the set : $$v+W=\{v+w\mid w \in W\} \subset V$$, called the coset of W containing v. Let $$V/W=\{v+W\mid v \in V\}$$ which we call the quotient space of $$V$$ modulo $$W$$.

Let T be a linear operator on a finite-dimensional vector space V, and suppose W is a T-invariant subspace of V. Let $$\bar{T}:V/W \rightarrow V/W$$ be a linear operator. Prove that the minimal polynomial of $$\bar{T}$$ divides the minimal polynomial of T.

My try: Let $$p(\bar{T})$$ be a polynomial of $$\bar{T}$$, then for any $$v \in V$$, $$(g(\bar{T}))(v+W)=g(T)(v)+W=0+W=W$$. Since it is T-invariant, by theorem( Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a T-invariant subspace of V, then the characteristic polynomial of $$T_W$$ divides the characteristic polynomial of T), we have proved the statement.

Is this right?

• If you have shown that the minimal polynomial of $U$ divides any polynomial $g(t)$ such that $g(U)$ is the zero map, then yes. But it would be good to invoke this result explicitly, rather than elide it as you do in both cases. Apr 20, 2020 at 21:05