Answer
See below
Work Step by Step
Let $A,B$ be the solutions of the given solutions.
$\Rightarrow (A+B)x\\
=Ax+Bx\\
=0+0\\
=0$
If we assume that $A+B$ is also the solutions to the given equation, then $A+B\in S$.
$\Rightarrow A+B$ is closed under addition.
Set of Rational numbers is closed under usual addition.
Because our scalars can come from set of real numbers
In particular, let $\lambda$ be a scalar.
$\Rightarrow (\lambda A)x
=\lambda (Ax)\\
=\lambda .0\\
=0$
Assume $\lambda. A$ is a solution to the given equation, then $\lambda .A \in S$.
Therefore $S$ is closed under addition.
