Answer
$(A_1)$ Holds
$(A_2)$ Fails
Work Step by Step
$(A_1)$ Let $x,y\in \mathbf{Q}$
$\Rightarrow x,y$ are rational numbers
And we know that sum of two rational numbers is rational number
$\Rightarrow x+y\in \mathbf{Q}$
Set of Rational numbers is closed under usual addition.
$(A_2)$ Because our scalars can come from set of real numbers
In Particular ,Let $\lambda =\sqrt{2}$
Let $x=\frac{1}{2}\in \mathbf{Q}$
$\Rightarrow \lambda\cdot x=\sqrt{2}.\large\frac{1}{2}$
$\Rightarrow \lambda\cdot x=\frac{1}{\sqrt{2}}$
But $\frac{1}{\sqrt{2}}$ is not a rational number
Therefore set of rational number is not closed with respect to scalar multiplication where scalars can come from set of real numbers
