Chapter 6 - Linear Transformations - 6.2 The Kernel and Range of a Linear Transformation - 6.2 Exercises - Page 312: 2

$\{(0,y,0):\text{y is real}\}$

The kernel of a linear map $T : A → B$, is the set of all elements $v$ in $A$ for which $T(v) = 0$, where $0$ is the zero vector in B. Hence, here we need $(x,0,z)=(0,0,0)$. Thus $x=0,y=y,z=0$, and thus the kernel is $\{(0,y,0):\text{y is real}\}$.