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Abstract Algebra

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Moderate

Module over Ring of Polynomials: Linear Map on a Vector Space

ABSALG-X1LSSD

Let $V$ be a real vector space of dimension 2 and let $T:V\rightarrow V$ be a linear transformation. Recall that this means:

$$T(\alpha \vec{v}+\beta\vec{w})=\alpha T(\vec{v})+\beta T(\vec{w}),\quad \vec{v}, \vec{w}\in V,\quad \alpha,\beta\in\mathbb{R}.$$
Let $\vec{v}\in V$ and suppose that $\mathcal{B}=\{\vec{v}, T(\vec{v})\}$ is a basis of $V$ (that is, $\vec{v}$ and $T(\vec{v})$ are linearly independent).

Endow $V$ with the $\mathbb{R}[x]$-module structure defined by: $p(x)\vec{v}:=p(T)\vec{v}$, for $\vec{v}\in V$, and $p(x)\in\mathbb{R}[x]$.

In particular $T^n$, $n\ge 1$, is $T$ composed with itself $n$ times.

Suppose that the $\mathbb{R}[x]$-module $V$, defined above, is isomorphic to the $\mathbb{R}[x]$-module given by the quotient $\mathbb{R}[x]/(x^2+x+1)$, where $(x^2+x+1)$ is the ideal generated by $x^2+x+1$ in $\mathbb{R}[x]$.

Which of the following is the matrix of $T$ with respect to the basis $\mathcal{B}$?

Assume we write elements of $V$ as column vectors with respect to $\mathcal{B}$, and that matrices act on column vectors from the left.

A

$\begin{pmatrix}\omega&0\cr0&{\overline{\omega}}\end{pmatrix}$, where $\omega^2+\omega+1=0$ and ${\overline{\omega}}$ is the complex conjugate of $\omega$.

B

$\begin{pmatrix}{\overline{\omega}}&0\cr0&\omega\end{pmatrix}$, where $\omega^2+\omega+1=0$ and ${\overline{\omega}}$ is the complex conjugate of $\omega$.

C

$\begin{pmatrix}0&-1\cr1&-1\end{pmatrix}$

D

$\begin{pmatrix}\;\;0&\;\;1\cr-1&-1\end{pmatrix}$

E

$\begin{pmatrix}0&\;\;1\cr1&-1\end{pmatrix}$