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

$\{a_1x+a_2x^2:\text{$a_1,a_2$ are 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. We have $a_0=0$ and the others are free variables. Thus the kernel is $\{a_1x+a_2x^2:\text{$a_1,a_2$ are real}\}$.