Q10Multiple correct4 Marks16 Mar 2025

Suppose $T:\mathbb{R}^3\to\mathbb{R}^3$ is a linear transformation. Let $M_1$ and $M_2$ denote the matrix representations of $T$ with respect to distinct bases for both domain and codomain $\beta_1$ and $\beta_2$, respectively. Choose the correct statements.

A

$\operatorname{rank}(M_1)=\operatorname{rank}(M_2)$

B

$\operatorname{Columnspace}(M_1)=\operatorname{Columnspace}(M_2)$

C

$\operatorname{Nullspace}(M_1)=\operatorname{Nullspace}(M_2)$

D

$\operatorname{trace}(M_1)=\operatorname{trace}(M_2)$

E

$\det(M_1)=\det(M_2)$