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**1. Show that T isn't a linear transformation and provide a suitable counterexample.**

##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}##

**2. The attempt at a solution**

##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}##

##T(\vec{v}) = T\left(\begin{bmatrix}0\\0 \end{bmatrix}\right) = \begin{bmatrix}0 - 1 \\ 0 + 1 \end{bmatrix} = \begin{bmatrix}-1 \\ 1 \end{bmatrix}##

Nonlinear transformation due to constants: ##T \begin{bmatrix}-1 \\ 1 \end{bmatrix}##.