Matrix and characteristic polynomial for a given linear transformation

#linear_algebra

Let $V = {a_{0} + a_{1} \sqrt[3]{2} + a_{2} \sqrt[3]{4} ∣ a_{0}, a_{1}, a_{2} \in Q} \subseteq R$ . This set is a vector space over $Q$ .

Verify $V$ is closed under product (using the usual product operation in $R$ ).
Let $T : V \to V$ be the linear transformation defined by $T (v) = (\sqrt[3]{2} + \sqrt[3]{4}) v$ . Find the matrix that represents $T$ with respect to the basis ${1, \sqrt[3]{2}, \sqrt[3]{4}}$ for $V$ .
Determine the characteristic polynomial for $T$ .

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Let $V=\{a_0+a_1\sqrt[3]{2}+a_2\sqrt[3]{4}\mid a_0, a_1, a_2\in {\bf Q}\}\subseteq {\bf R}$. This set is a vector space over ${\bf Q}$.
\begin{enumerate}[label=\alph*)]
	\item Verify $V$ is closed under product (using the usual product operation in ${\bf R}$).
	\item Let $T:V\to V$ be the linear transformation defined by $T(v)=(\sqrt[3]{2}+\sqrt[3]{4}) v$. Find the matrix that represents $T$ with respect to the basis $\{1,\sqrt[3]{2},\sqrt[3]{4}\}$ for $V$.
	\item Determine the characteristic polynomial for $T$.
\end{enumerate}