Let \[A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}
\]

and Define \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(x) = A x \).

Find the image under \[
T \left( u = \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \, v = \begin{bmatrix} u \\ u \end{bmatrix} \right)
\]

b) Prove that a map \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T(x,y) = (x,y) \) is linear.