Understanding T-conductor in Linear Algebra I'm studying Hoffman/Kunze Linear Algebra 2E and I got stuck in understanding the concept of T-conductor. According to the book, T-conductor is defined as below: **Definition.** Let $W$ be an invariant subspace for $T$ and let $\alpha$ be a vector in $V$. The $T$-conductor of $\alpha$ into $W$ is the set $S_T(\alpha;W)$, which consists of all polynomials $g$ (over the scalar field) such that $g(T)\alpha$ is in $W$. (p.201, Sec. 6.4) My question is that isn't the $T$-conductor same as the set of all polynomials over given field? If $W$ is invariant under $T$, then $W$ is invariant under every polynomial in $T$ and this is how I came up with this question. If I'm right, why did the textbook defined $T$-conductor and $F[x]$ as if they are different things? I'm so confused. Can somebody tell me what I am missing?

The vector $\alpha$ belongs to $V$. Your remark would be correct if we were assuming that $\alpha\in W$. In particular, if $\alpha\in W$, $S_T(\alpha;W)$ is indeed the set of all polynomials.