Thread: solution space

Is only a homogeneous system may have a solution space?
Thanks in advance...

Yes, because the set of solutions of a non-homogeneous system is not closed under addition and, therefore, is not a vector space.

Specifically, if u and v satisfy Ax= 0, then A(au+ bv)= aA(u)+ bA(v)= 0 but they satisfy Ax= p, where p is not 0, then A(au+ bv)= ap+ bp= (a+ b)p which is not, in general, equal to p.