How Do We Prove That 'P' Doesn't Vary for Different Free Variables?

[mathworker]

I am studying Introduction to linear Algebra by Gilbert Strang while calculating the particular solution P for $Ax=b$,he made the free variables $0$ to calculate the particular solution and said that P along with linear combinations of null space solutions make up the complete set.

I understood that particular solution is 'particular' because there is only one solution to system of equation but how do we prove that 'P' doesn't vary for different free variable ( I mean other than $0$)??​

 

[Evgeny.Makarov]

Re: particular sollution for Ax=b

how do we prove that 'P' doesn't vary for different free variable ( I mean other than $0$)??

A particular solution definitely varies with the choice of values of free variables because these values are a part of a solution. But the set of all solutions to the system does not depend on the choice of a particular solution.

There is the following fact. Let $L$ be a linear subspace and let $v$ be a particular vector (not necesarily in $L$). If $v'\in L+v$, then $L+v=L+v'$. Indeed, suppose that $v'=u+v$ where $u\in L$. Then an arbitrary vector in $L+v$ has the form $w+v$ where $w\in L$. But $w+v=w+v+u-u=(w-u)+v'$ and $w-u\in L$. The fact that $L+v'\subseteq L+v$ is proved similarly.

Returning to the system of linear equations, $L$ is the null space of $A$ and $v,v'$ are particular solutions. The choice of a particular solution does not change the set of all solutions.