Solution sets of Linear Systems Theorem

Theorem

The solution set $S$ of $Ax=b$ is a Affine Subset $x_p+ker(T_A)$ Formally, all solutions can be written as $x=x_p+x_h$ where:

• $x_p$ is a particular solution s.t $A{x}_p=b$
• $x_h$ is a homogeneous solution of $A{x}_h=0$
• If $Ax=b$ has no solution, then it is inconsistent and has $\varnothing$ as the solution set

Proof

1. Suppose that $\overrightarrow{x}=x_p+x_k$
2. We get $Ax=A(x_p+x_h)$
3. $=A{x}_p+A{x}_n$ by matrix multiplication distributive prop
4. $=b+0$
5. $=b$
6. Suppose that $\overrightarrow{x}$ is any solution of $Ax=b$
7. We know $Ax=b$ and $A{x}_p=b$
8. This gives us: $Ax-A{x}_p=b-b=0$
9. $⟺A(x-x_p)=0$
10. Therefore, $x-x_p$ is a soln of the homogeneous system $Ax=0$
11. Thus, we get $x_n=x-x_p⟺x=x_n+x_p$