How to show that a given transformation is linear from $P_2$ to a $2 \times 3$ Matrix?

Question

I'm given the following definition for a transformation

$ T: \mathbb{P_2}\rightarrow M_{2 \times 3}\space \space by \space\space T(ax^2 + bx +c) = \begin{bmatrix}a+c &b && 0\\b & 2b && 0\\\end{bmatrix}$

And I need to show that it is a linear transformation. I have done this in the past for more basic transformations, but since we are dealing with a vector space of polynomials I am not sure how to apply the definition of a linear transformation.