TITLE: Is this a linear transformation?
QUESTION [6 upvotes]: Let $T$ be a transformation from $P_2$ to $P_2$ (where $P_2$ is the space of all polynomials with degree less than or equal to $2$)
$$T(f(t)) = f''(t)f(t)$$
I'm tempted to say that this is not a linear transformation because
$$T(f(t) + g(t)) = (f''(t) + g''(t))(f(t) + g(t))$$
Which does not equal
$$T(f(t)) + T(g(t))$$
But I'm not sure if I did that correctly...

REPLY [3 votes]: If $f(t)=at^2+bt+c$, then $T(f)(t)=2a^2t^2 +2abt+2ac$. The $a^2$ in the quadratic term looks suspicuously non-linear. In fact, if $f(t)=t^2$, then $T(f)(t) = 2t^2$ and $T(-f)(t)=2t^2$ as well. Assuming $2\ne0$, this contradicts linearity as that requires $T(f)+T(-f)=0$.
On the other hand, if $2=0$ in your base fiedl, then $T$ is the zero map and trivially linear.