Chapter 6 - Linear Transformations - 6.2 The Kernel and Range of a Linear Transformation - 6.2 Exercises - Page 312: 8

$\{a_0:\text{$a_0$ 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 $a_1+2a_2x+3a_3x^2=0$, so $a_1=a_2=a_3=0$, and the others are free variables. Thus the kernel is $\{a_0:\text{$a_0$ is real}\}$.