For some reason, this particular problem is throwing me off:
Find the kernel of the linear transformation:
$T: P_2 \rightarrow P_1$
$T(a_0+a_1x+a_2x^2)=a_1+2a_2x$
Since the kernel is the set of all vectors in $V$ that satisfy $T(\vec{v})=\vec{0}$, it's obvious that $a_0$ can be any real number. What matters, if I understand correctly, is that $a_1$ and $a_2$ should equal 0 in order to satisfy the zero vector (i.e. $0+2(0)x$).
Granted that what I stated is correct, why would my book say that the $\ker(T)=\{a_0: a_0 \; \text{is real}\}$?
Yes, $a_0$ can be any real number, but what must $a_1$ or $a_2$ equal? I don't see it specified within the set. Perhaps it's implied - I'm not sure.
Let me add some more detail:
Again, if I understand correctly, I could make a system of equations as such:
$a_1 = 0$
$2a_2 = 0$
From that I can translate it into a matrix and find that $a_1$ and $a_2$ really does equal zero.