Suppose that a single equation from this system (the $i$-th one) has the form,
\begin{equation*}
a_{i1}x_1+a_{i2}x_2+a_{i3}x_3+\dots+a_{in}x_n=0
\end{equation*}
Evaluate the left-hand side of this equation with the components of the proposed solution vector $\vect{v}$,
\begin{align*}
a_{i1}\left(4u_1\right)&+a_{i2}\left(4u_2\right)+a_{i3}\left(4u_3\right)+\dots+a_{in}\left(4u_n\right)\\
&=4a_{i1}u_1+4a_{i2}u_2+4a_{i3}u_3+\dots+4a_{in}u_n&&\text{Commutativity}\\
&=4\left(a_{i1}u_1+a_{i2}u_2+a_{i3}u_3+\dots+a_{in}u_n\right)&&\text{Distributivity}\\
&=4(0)&&\text{$\vect{u}$ solution to $\homosystem{A}$}\\
&=0
\end{align*}
So $\vect{v}$ makes each equation true, and so is a solution to the system.
Notice that this result is not true if we change $\homosystem{A}$ from a homogeneous system to a non-homogeneous system. Can you create an example of a (non-homogeneous) system with a solution $\vect{u}$ such that $\vect{v}$ is not a solution?