1. ## Linear transformation

φ : P4 ->R given by φ(p(x)) = p′′(0), where p′′(x) denotes the second derivative.

Is the linear transformation injective, surjective?

2. Originally Posted by skittle
φ : P4 ->R given by φ(p(x)) = p′′(0), where p′′(x) denotes the second derivative.

Is the linear transformation injective, surjective?

Hint: $\phi$ is injective iff $\ker\phi=\{0\}$ , and it is surjective iff every real number

can be written as $p''(0)$ , for some polynomial $p(x)$ of degree at most 4...

Tonio