# Proof for Solution set of the matrix equation Ax=b

The following theorem is given in David C Lay's Linear Algebra and its applications book

Suppose the equation $A$x=b is consistent for some given b, and let p be a solution.Then the solution set of $A$x=b is the set of all vectors of the form $\;$w=p + $v_h$,$\;$where $v_h$ is any solution of the homogeneous equation Ax=$0$

I can prove why $\;$w=p + $v_h$ is a solution for the matrix equation $A$x=b, but how can we prove the solution set only contains all vectors of this form,i mean any solution of $A$x=b in fact is of this certain form.

Hope i was able to convey my question clearly, Thanks for the help

• Why do you call it matrix equation if the unknown is a vector, not a matrix? – Roland Apr 25 '17 at 20:53

Let $w$ a solution of $Ax=b$. Then, $A(p-w)=Ap-Aw=b-b=0$. So, $w=p+(w-p)$ is the decomposition required.