Chapter 6 - Linear Transformations - 6.2 The Kernel and Range of a Linear Transformation - 6.2 Exercises - Page 312: 9

$\{(0,0)\}$.

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 $y-x=0\\x=y$ and $x+2y=0\\x+2x=0\\3x=0\\x=0$ and thus $y=0$, so $x=y=0$. Thus the kernel is $\{(0,0)\}$.