If $x, y \in \ker(A)$ for some matrix $A$, how to show that any vector of the form $ax+by$, where $a,b \in \mathbb R$ is also in $\ker(A)$?

Question

Would my reasoning below be correct?

$Ax=0$ and $Ay=0$, so $A(ax)=0$ and $B(by)=0$, so $A(ax+by)=0$. Hence $ax+by$ is also in $\ker(A)$.

Answer 1

I miss the reasoning for $ A(ax+by)=0$ !

$ A(ax+by)=A(ax)+A(by)=aA(x)+bA(y)=a0+b0=0$.