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Let $T:V \longrightarrow V$ a linear operator, where $V$ is a vector space over the field $\mathbb{K}$. Show that if $p(x),q(x)\in \mathcal{P}(\mathbb{K})$, then

$$(p\cdot q)(T)(v)=p(q(T))(v), \ \ \forall v\in V.$$

I have some examples but I get no such equality. I am taking $p \cdot q $ as the product of polynomials, as in the book i'm following do not indicate the meaning of that product. Could anyone help me or give me some pointers?

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1 Answer 1

You might already know this, but as indicated by $p(q(T))$, $p\cdot q$ is related to a Function composition of $q$ and $p$:

First apply $q$ on $x$, then $p$ on $q(T)$, so to me, this looks just like a definition.

But, it's also true in a multiplicative sense, if $p(x)$ or $q(x)$ is constant :-).

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