Proof of Theorem PSPHS

Let $\vectorlist{A}{n}$ be the columns of the coefficient matrix $A$.

(⇐) Suppose $\vect{y}=\vect{w}+\vect{z}$ and $\vect{z}\in\nsp{A}$. Then \begin{align*} \vect{b} &= \vectorentry{\vect{w}}{1}\vect{A}_1+ \vectorentry{\vect{w}}{2}\vect{A}_2+ \vectorentry{\vect{w}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{w}}{n}\vect{A}_n &&\knowl{./knowls/theorem.SLSLC.knowl}{\text{Theorem SLSLC}}\\ &= \vectorentry{\vect{w}}{1}\vect{A}_1+ \vectorentry{\vect{w}}{2}\vect{A}_2+ \vectorentry{\vect{w}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{w}}{n}\vect{A}_n +\zerovector &&\knowl{./knowls/property.ZC.knowl}{\text{Property ZC}}\\ &= \vectorentry{\vect{w}}{1}\vect{A}_1+ \vectorentry{\vect{w}}{2}\vect{A}_2+ \vectorentry{\vect{w}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{w}}{n}\vect{A}_n &&\knowl{./knowls/theorem.SLSLC.knowl}{\text{Theorem SLSLC}}\\ &\quad\quad + \vectorentry{\vect{z}}{1}\vect{A}_1+ \vectorentry{\vect{z}}{2}\vect{A}_2+ \vectorentry{\vect{z}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{z}}{n}\vect{A}_n\\ &= \left(\vectorentry{\vect{w}}{1}+\vectorentry{\vect{z}}{1}\right)\vect{A}_1+ \left(\vectorentry{\vect{w}}{2}+\vectorentry{\vect{z}}{2}\right)\vect{A}_2 &&\knowl{./knowls/theorem.VSPCV.knowl}{\text{Theorem VSPCV}}\\ &\quad\quad \left(\vectorentry{\vect{w}}{3}+\vectorentry{\vect{z}}{3}\right)\vect{A}_3+ \cdots+ \left(\vectorentry{\vect{w}}{n}+\vectorentry{\vect{z}}{n}\right)\vect{A}_n\\ &= +\vectorentry{\vect{w}+\vect{z}}{1}\vect{A}_1+ \vectorentry{\vect{w}+\vect{z}}{2}\vect{A}_2+ \cdots+ \vectorentry{\vect{w}+\vect{z}}{n}\vect{A}_n &&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &= \vectorentry{\vect{y}}{1}\vect{A}_1+ \vectorentry{\vect{y}}{2}\vect{A}_2+ \vectorentry{\vect{y}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{y}}{n}\vect{A}_n &&\text{Definition of $\vect{y}$} \end{align*} Applying Theorem SLSLC we see that the vector $\vect{y}$ is a solution to $\linearsystem{A}{\vect{b}}$.

(⇒) Suppose $\vect{y}$ is a solution to $\linearsystem{A}{\vect{b}}$. Then \begin{align*} \zerovector &= \vect{b}-\vect{b}&&\text{}\\ &= \vectorentry{\vect{y}}{1}\vect{A}_1+ \vectorentry{\vect{y}}{2}\vect{A}_2+ \vectorentry{\vect{y}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{y}}{n}\vect{A}_n &&\knowl{./knowls/theorem.SLSLC.knowl}{\text{Theorem SLSLC}}\\ &\quad\quad - \left( \vectorentry{\vect{w}}{1}\vect{A}_1+ \vectorentry{\vect{w}}{2}\vect{A}_2+ \vectorentry{\vect{w}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{w}}{n}\vect{A}_n \right)\\ &= \left(\vectorentry{\vect{y}}{1}-\vectorentry{\vect{w}}{1}\right)\vect{A}_1+ \left(\vectorentry{\vect{y}}{2}-\vectorentry{\vect{w}}{2}\right)\vect{A}_2 &&\knowl{./knowls/theorem.VSPCV.knowl}{\text{Theorem VSPCV}}\\ &\quad\quad + \left(\vectorentry{\vect{y}}{3}-\vectorentry{\vect{w}}{3}\right)\vect{A}_3+ \cdots+ \left(\vectorentry{\vect{y}}{n}-\vectorentry{\vect{w}}{n}\right)\vect{A}_n\\ &= \vectorentry{\vect{y}-\vect{w}}{1}\vect{A}_1+ \vectorentry{\vect{y}-\vect{w}}{2}\vect{A}_2 &&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &\quad\quad +\vectorentry{\vect{y}-\vect{w}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{y}-\vect{w}}{n}\vect{A}_n \end{align*} By Theorem SLSLC we see that the vector $\vect{y}-\vect{w}$ is a solution to the homogeneous system $\homosystem{A}$ and by Definition NSM, $\vect{y}-\vect{w}\in\nsp{A}$. In other words, $\vect{y}-\vect{w}=\vect{z}$ for some vector $\vect{z}\in\nsp{A}$. Rewritten, this is $\vect{y}=\vect{w}+\vect{z}$, as desired.