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

  • $\begingroup$ Why do you call it matrix equation if the unknown is a vector, not a matrix? $\endgroup$ – 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.

  • $\begingroup$ Thanks the words " decomposition required" helped me better understand the argument. so any solution can be decomposed into this form particular solution + w-p (solution to corresponding homogeneous system) $\endgroup$ – sarat Apr 25 '17 at 20:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.