Linear System Solutions and the Role of Scalar Multiplication

[golriz]

I have a question:
If the vectors of v & u are solutions of the nonhomogeneous linear system Ax=b, then ru+sv is a solution of the nonhomogeneous system for any real values of r and s.
is this statement true?
is this statement true for homogeneous systems too?

 

[konthelion]

Hint: When a nonhomogeneous linear system has infinite solutions, then the general solutions can be written in parametric vector form.

This can be used to answer both questions.

 

[Defennder]

You should be able to find a counterexample easily for the first question. For the second, try writing the matrix equation for Ax=b using ru+sv.