Linear Algebra

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Difficult

Permutation Character

LINALG-CXGVGN

If $\mathcal{E}=\{\vec{e}_1,\vec{e}_2,\vec{e}_3,\vec{e}_4\}$ is the standard ordered basis for $\mathbb{R}^4$ and $\mathcal{E}'$ is an ordered basis obtained by rearranging the vectors in $\mathcal{E}$ and if $P$ is the $4\times 4$ matrix such that
$$P[\vec{x}]_{\mathcal{E}'}=[\vec{x}]_{\mathcal{E}}$$
then for which $\mathcal{E}'$ is $tr(P)=0$. Select all correct choices.

A

$\mathcal{E}'=\{\vec{e}_2,\vec{e}_3,\vec{e}_4,\vec{e}_1\}$

B

$\mathcal{E}'=\{\vec{e}_2,\vec{e}_1,\vec{e}_3,\vec{e}_4\}$

C

$\mathcal{E}'=\{\vec{e}_2,\vec{e}_1,\vec{e}_4,\vec{e}_3\}$

D

$\mathcal{E}'=\{\vec{e}_2,\vec{e}_3,\vec{e}_1,\vec{e}_4\}$

E

$\mathcal{E}'=\{\vec{e}_4,\vec{e}_3,\vec{e}_2,\vec{e}_1\}$

F

$\mathcal{E}'=\{\vec{e}_1,\vec{e}_2,\vec{e}_3,\vec{e}_4\}$