AMU 2025 — Mathematics PYQ

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^3\times \mathbb{R}$ be a linear transformation defined by  

$T(x_1,x_2,x_3,x_4)=(x_1-x_2,x_2-x_3,x_3-x_4)$

Then which of the following is true?

(i) $\dim (\ker T)=1$ if $C\ne 0$  

(ii) $\dim (\ker T)=0$ if $C=0$  

(iii) $\dim (\ker T)=1$ if $T$ is onto