Linear transformation from a vector space of polynomials

#linear_algebra

Let $P_{3}$ be the real vector space of all real polynomials of degree three or less. Define $L : P_{3} \to P_{3}$ by $L (p (x)) = p (x) + p (- x)$ .

Prove $L$ is a linear transformation.
Find a basis for the null space of $L$ .
Compute the dimension of the image of $L$ .

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Let $P_3$ be the real vector space of all real polynomials of degree three or less. Define $L:P_3\to P_3$ by $L(p(x))=p(x)+p(-x)$.
\begin{enumerate}[label=\alph*)]
	\item Prove $L$ is a linear transformation.
	\item Find a basis for the null space of $L$.
	\item Compute the dimension of the image of $L$.
\end{enumerate}