The Complete Solution to the matrix equation Ax = b

[rtareen]

TL;DR Summary

We are trying to find the complete solution to the matrix equation Ax = b where A in an m x n matrix and m is not equal to n.

We are trying to find the complete solution to the matrix equation $A\vec x = \vec b$ where A is an m x n matrix and $\vec b$ can be anything except the zero vector. The entire solution is said to be:
$\vec x = \vec x_p + \vec x_n$
where $\vec x_p$ is the solution for a particular $\vec b$ and $\vec x_n$ is the entire nullspace.

I don't understand this. Why is the nullspace included in the solution, when it is defined to be the solution when $\vec b = \vec 0$? Or else what is this equation really saying?

 

[Abhishek11235]

We have for any x:
$Ax= A(x_n+x_p)= b$
Note that $x_n$ can always be added to the solution since $Ax_n=0$ and $Ax_p=b$

 

[fresh_42]

If we have two particular solutions $\vec{x}_p$ and $\vec{x}_q$ then $A\vec{x}_p-A\vec{x}_q=\vec{b}-\vec{b}=\vec{0}$. Thus $\vec{x}_p-\vec{x}_q \in \operatorname{null}(A)$. Now
$A\vec{x}_p =\vec{b}=A\vec{x}_q + A(\vec{x}_p-\vec{x}_q)=A\vec{x}_p+A\vec{x}_n.$

On the other hand, if we start with any $\vec{x}_n\in \operatorname{null}(A)$, and have one particular solution $\vec{x}_p,$ then $A(\vec{x}_p+\vec{x}_n)=\vec{b}+\vec{0}=\vec{b}.$ So we may add any vector from the nullspace and get another particular solution $\vec{x}_p+\vec{x}_n.$

This mean that all vectors in the set $\vec{x}_p +\operatorname{null}(A)$ are all possible solutions to $A\vec{x}=\vec{b}.$

 

[rtareen]

These answers make sense. Thanks guys.