So I have the following question here:
Consider a linear system of $m$ equations in $n$ variables. Let A be its matrix of coefficients. Then that system can be written in matrix form as $$A\mathbf{x}=\mathbf{b}.$$ Suppose that $s_1$ is a solution of that system of equations. Let $\mathbf{s}$ be any solution of $A\mathbf{x}=\mathbf{b}$. Explain why $\mathbf{s}$ is equal to $\mathbf{s_1}+\mathbf{s}_{0}$ where $\mathbf{s}_0$ is a solution of the homogenous system $A\mathbf{x}=\mathbf{0}$.
Okay so from what I can see, this means that $A\mathbf{s_1}=b$ and $A\mathbf{s}=b$ must hold I suppose?
If $s_0$ is a solution to $Ax=0$ that means $As_0=0$ I guess? Since we have $A\mathbf{s_1}=b$ and $A\mathbf{s}=b$ that means we can probably do $As_1=As$ or $As-As_1=0$ which implies $A(s-s_1)=0$. Since $As_0=0$, we have $A(s-s_1)=As_0$ meaning that $s-s_1=s_0$ or $s=s_0+s_1$.
Is that the right idea?