(Solved): Determine whether the following transformations are linear. If a transformation is linear, provide ...

Determine whether the following transformations are linear. If a transformation is linear, provide a proof that it is linear by verifying that \( (\mathrm{LT} 1) \) and \( (\mathrm{LT} 2) \) hold. (See the definition of linear transformation.) If the transformation is not linear, state one of the properties of a linear transformation that does not hold - either (LT1), (LT2), or the \( \mathbf{0} \) test (see Q5 in Linear Transformations for the 0 test) - and give a counter-example showing that the property fails. (a) Let \( T: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3} \) be given by \[ T\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{c} z \\ \cos (\theta) x-\sin (\theta) y \\ \sin (\theta) x+\cos (\theta) y \end{array}\right] \] where \( \theta \) is some fixed real number. (b) Let \( T: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{2} \) be given by \[ T\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{c} 2 x+y-4 z \\ 3 x+x y z \end{array}\right] \]