Chapter 6 - Linear Transformations - 6.2 The Kernel and Range of a Linear Transformation - 6.2 Exercises - Page 312: 10

$\{(x,x):\text{x is real}\}$.

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 $y-x=0\\x=y$ and $x-y=0\\x-x=0$, which is always true. Thus, $x=y$. Thus the kernel is $\{(x,x):\text{x is real}\}$.