How to calculate subspace of a set of solutions of matrix Ax=b

Question

I am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question:

I think i understand how to check whether vectors are within a subspace R, but how would i calculate this?

Thanks a lot, really hope you can help me out!

Answer 1

The set of solutions of $A\mathbf{x}=\mathbf{b}$ is not a subespace of $\mathbb{R}^4$ becasuse the null vector $\mathbf{0}=\begin{bmatrix}{0}\\{0}\\{0}\\{0}\end{bmatrix}$ does not satisfy $A\mathbf{0}=\mathbf{b}$.