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Let $S$ be the subspace formed by $e^x$,$xe^x$, and $x^2e^x$. Let $D$ be the differentiation operator on $S$. Prove that $D$ is a linear transformation.

I am not sure about my setup and confused with my result for scalar multiplication.

$T(0) = 0 \frac{d}{dx} = 0$

Let $u = a_0e^x + a_1xe^x+a_2x^2e^x$, $v= b_0e^x + b_1xe^x+b_2x^2e^x$.

Then, $T(u+v) = (a_0+b_0)e^x\frac{d}{dx} + (a_1+b_1)xe^x\frac{d}{dx} +(a_2+b_2)x^2e^x$

$\Rightarrow$.....$\Rightarrow$ $(a_0+a_1)e^x + (a_1+2a_2)xe^x + (a_2)x^2e^x + (b_0+b_1)e^x$...

$=T(u)+T(v)$

Then for, $T(cu) = ce^x\frac{d}{dx} + cxe^x\frac{d}{dx} + cx^2e^x\frac{d}{dx}$

we get, $\Rightarrow 2ce^x+ 3cxe^x + cx^2e^x$ ?

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    You've got the right idea, but the presentation can be more clear. – 2017-02-24

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Scalar multiplication....

you need to show that $T(c\mathbf u) = cT(\mathbf u)$

$\mathbf u = a_0 e^x + a_1 x e^x + a_2 x^2 e^x$

$T(\mathbf u) = (a_0 + a_1) e^x + (a_1 + 2a_2) x e^x + a_2 e^x\\ T(c\mathbf u) = (ca_0 + ca_1) e^x + (ca_1 + 2ca_2) x e^x + ca_2 e^x = cT(\mathbf u)$