# Thread: Proving that something is a subspace

1. ## Proving that something is a subspace

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 substituting x = 0 into 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.

2. Let $\displaystyle b_1,b_2 \in S$ so $\displaystyle b_1+b_2=Ax_1 + Ax_2 = A(x_1+x_2)$ so that $\displaystyle x_1+x_2 \in \mathbb{R}^3$.