Q40Multiple correct2 Marks6 Aug 2023

Consider $V=\left\{\begin{bmatrix}a&b\c\&a\end{bmatrix}:c=a+b;\ a,b,c\in\mathbb{R}\right\}$ and $T:V\to\mathbb{R}^4$ defined by $T(A)=(a,b,c,a+b-c)$. Choose the correct options.

A

$T$ is onto but not one-one.

B

$T$ is one-one but not onto.

C

The nullspace of $T$ is a $2$-dimensional subspace of $V$.

D

The range of $T$ is a $2$-dimensional subspace of $\mathbb{R}^4$.