φ : P4 ->R given by φ(p(x)) = p′′(0), where p′′(x) denotes the second derivative. Is the linear transformation injective, surjective?
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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: $\displaystyle \phi$ is injective iff $\displaystyle \ker\phi=\{0\}$ , and it is surjective iff every real number can be written as $\displaystyle p''(0)$ , for some polynomial $\displaystyle p(x)$ of degree at most 4... Tonio
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