Q1Single correct6 Marks10 May 2026

Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be a linear transformation. Let $B_1 = \{(1, 0, 1), (1, 1, 0), (0, 1, 1)\}$ and $B_2 = \{(1, 1), (1, -1)\}$ be ordered bases for the domain and co-domain respectively. Suppose that the matrix representation of $T$ with respect to $B_1$ and $B_2$ is $[T]_{B_1}^{B_2} = \begin{bmatrix} 0 & -1 & 2 \\ -1 & 2 & 0 \end{bmatrix}$. Which of the following defines the transformation $T$?

A

T(x, y, z) = (x - 2y, 2x + y - 3z)

B

T(x, y, z) = (-x - 2y, -2x - y + 3z)

C

T(x, y, z) = (2x - y, -x + 2y + 3z)

D

T(x, y, z) = (-x + 2y, -2x - y + 3z)