You can treat any polynomial $P_n$ space as an $R^n$ space.
That is, in your case the polynomial is $P_2$ and it can be converted to $R^3$.
The logic is simple, each coefficient of the term in the polynomial is converted to a number in $R^3$.
In the end of this conversion you'll get an isomorphics spaces/subspaces.
In your case :
The polynomial : $a + bx + cx^2$ can be converted to the cartesian product: $(c, b, a)$
I chose the coefficient from the highest degree to the smallest. That is important only to set some ground rules so you will know how to revert your cartesian product into polynomial again, if you want to do so. *You can choose any other order.
Now because these are isomorphics subspaces you'll get the same Kernel and the same Image.
I hope It helps.