The question:

Show that the set S = {$\displaystyle b \in \mathbb{R}^2 : b = Ax \textrm{ for some } x \in \mathbb{R}^3$} where A = $\displaystyle \[ \left( \begin{array}{ccc}

2 & -3 & 1 \\

4 & 5 & -3 \end{array} \right)\]$ is a subspace of $\displaystyle \mathbb{R}^2.$ Explain why each column of the matrix belongs to the set S.

I'm having trouble with this one. I recognize the Ax=b form, but I'm not sure how to prove that it is a subspace. I know the Subspace Theorem, and I managed to show that it isn't an empty-set by substitutingx=0into the matrix. But when it comes to closure under vector addition, I'm getting confused. Could someone shed some light on how I should attempt this? Thanks.