## Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis

## Problem 153

Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.

Let $W$ be the following subset of $P_3$.

\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and $p^{\prime\prime}(x)$ is the second derivative of $p(x)$.

Show that $W$ is a subspace of $P_3$ and find a basis for $W$.

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