Let $\mathcal{P}_n$ denote the vector space of polynomials in the variable $x$ of degree $n$ or less with real coefficients. Let $D : \mathcal{P}_3 \to \mathcal{P}_2$ be the function that sends a polynomial to its derivative. That is, $D(p(x)) = p'(x)$ for all polynomials $p(x) \in \mathcal{P}_3$. Is $D$ a linear transformation?

Let $p(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0$ and $q(x) = b_3 x^3 + b_2 x^2 + b_1 x + b_0$ be any two polynomials in $\mathcal{P}_3$ and $c \in \mathbb{R}$.

1. $D(p(x)+q(x)) =$ . (Enter $a_3$ as $\verb+a3+$, etc.)
   $D(p(x)) + D(q(x)) =$ $+$ .
   Does $D(p(x)+q(x)) = D(p(x)) + D(q(x))$ for all $p(x), q(x) \in \mathcal{P}_3$? choose Yes, they are equal No, they are not equal
   
   
2. $D(c p(x)) =$ .
   $c(D(p(x))) =$ $\Big($ $\Big)$.
   Does $D(c p(x)) = c(D(p(x)))$ for all $c \in \mathbb{R}$ and all $p(x) \in \mathcal{P}_3$? choose Yes, they are equal No, they are not equal
   
   
3. Is $D$ a linear transformation? choose f is a linear transformation f is not a linear transformation