Left and Right Inverses

Due by Mon, 05 Feb 2018

Fix $n > 0$ and let $V$ be the vector space of polynomials $p(x)$ over $\mathbb{R}$ of degree at most $n$. Let $L: V \to V$ be the linear tranformation $L(p) = \frac{dp}{dx}$, i.e., $L$ maps a polynomial $p$ to its derivative $\frac{dp}{dx}$.

  1. Prove or disprove: $L$ is injective.
  2. Prove or disprove: $L$ is surjective.
  3. Either describe a left inverse for $L$ (and prove that it is a left inverse) or prove that $L$ has no left inverse.
  4. Either describe a right inverse for $L$ (and prove it is a right inverse) or prove that $L$ has no right inverse.

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