A transformation $T: V \mapsto W$ is linear $\iff$ $T(cx + y) = cT(x) + T(y)$ where $x,y \in V$

Since $V$ is defined as $\text{span}(A)$, we can choose two arbitrary elements of $V$ to be $a_{1}A$ and $a_{2}A$ where $a_1 , a_2 \in \mathbb{R}$

Now, we try to prove:

$T(c(a_{1}A) + a_{2}A) = cT(a_{1}A) + T(a_{2}A)$

We can simplify this to make it easier to work with:

$T(c(a_{1}A) + a_{2}A) = T(c a_1 A + a_2 A) = T((ca_1 + a_2) A)$

$T((ca_1 + a_2) A) = T((ca_1 + a_2)e^x , (ca_1 + a_2) \sin{x} , (ca_1 + a_2) e^x \cos{x} , (ca_1 + a_2) \sin{x} , (ca_1 + a_2) \cos{x}$

$ = \left[ (ca_1 + a_2)e^x , (ca_1 + a_2) \cos{x} , (ca_1 + a_2) (e^x \cos{x} - e^x \sin{x}) , (ca_1 + a_2) \cos{x} , -(ca_1 + a_2) \sin{x} \right]$

$= \left[ ca_1 e^x + a_2 e^x , ca_1 \cos{x} + a_2 \cos{x}, \dots \right]$

$=c\left[a_1 e^x + a_2 e^x , a_1 \cos{x} + a_2 \cos{x} , \dots \right]$

$=cT(a_1 A) + T(a_2 A)$

Since we have that $T(c(a_{1}A) + a_{2}A) = cT(a_1 A) + T(a_2 A)$, we have that $T$ is linear.