Q3Multiple correct4 Marks6 Apr 2026

Let $T:\mathbb{R}^3\to\mathbb{R}^3$ be defined by $T(x,y,z)=(x+y,\ y+z,\ z+x+ky)$, where $k\in\mathbb{R}$. Which statements are true?

A

$T$ is injective if and only if $k\neq -1$.

B

$\dim(\ker(T))=1$ when $k=-2$.

C

The image of $T$ is a plane in $\mathbb{R}^3$ for all $k$.

D

The vector $(1,-1,1)$ is in $\ker(T)$ for some $k$.

E

There does not exist a non-zero vector $v$ such that $T(v)=v$.