Solutions of Nonhomogeneous Systems

[HelpNeeder]

Can the particular solution p also have free variables or is p one unique solution?

And if I'm right, whenever Ax = b has many solutions, you have the free choice to choose a particular solution out of the many solutions.
However, what I do not understand is that each time you choose a different p, the set of all solutions w changes (because p changes and w = p + v_h). So then you get different w's, which means you get different sets of solutions for the same equation Ax = b?

 

[lex]

Just because you use a different [MATH][/MATH][MATH]p[I][/I][/MATH] doesn't mean that the SET of solutions formed by all combinations [MATH]p+v_h[/MATH] will not be the same, (even though for a specific [MATH]v_h[/MATH], [MATH]\quad w_1=p_1+v_h[/MATH] will not be the same as [MATH]w=p+v_h[/MATH].
(Just as the lines generated by [MATH]\left(\begin{matrix}3\\2\\7\\\end{matrix}\right)+\lambda\left(\begin{matrix}4\\8\\2\\\end{matrix}\right)[/MATH]and by [MATH]\left(\begin{matrix}7\\10\\9\\\end{matrix}\right)+\lambda\left(\begin{matrix}-2\\-4\\-1\\\end{matrix}\right)[/MATH] are the same. I.e. they comprise the same SET of points, even if they are generated differently - the same value of lambda will generate different points).

 

[HelpNeeder]

I see, thank you!