$T: P_2 \to P_2$ defined by $T(p(x)) = p(x) + p(-x)$. Find basis for $\ker T$
Here is my solution: $p(x) = ax^2 + bx + c$ $p(x) + p(-x) = 0 \to 2ax^2+2c =0$ So, a = c = 0.
So, basis of $\ker T$ is $x$.
The thing that I'm not sure is: because $2ax^2 + 2c = 0$ with all x, so $a = c = 0$. Can I say that?
Thanks :)