Exercise 10.1.1

Consider the following functions $T:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$. For each of these functions $T$, show that it is not linear either by showing that $T$ does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector.

1. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1\\ 2y-3x+z\end{array}\right]$.

2. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y^{2}+3z\\ 2y+3x+z\end{array}\right]$.

3. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z\\ 2y+3x+z\end{array}\right]$.