\(\lambda(\vec{u} + \vec{v}) = \lambda\vec{u} + \lambda\vec{v}\)

$$\begin{align}
\lambda(\vec{u} + \vec{v})
&= \lambda \left(\begin{bmatrix}u_1 \\ \vdots \\ u_n\end{bmatrix} + \begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix}\right) \\[1em]
&= \lambda \begin{bmatrix}u_1 + v_1 \\ \vdots \\ u_n + v_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}\lambda(u_1 + v_1) \\ \vdots \\ \lambda(u_n + v_n)\end{bmatrix} \\[1em]
&= \begin{bmatrix}\lambda u_1 + \lambda v_1 \\ \vdots \\ \lambda u_n + \lambda v_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}\lambda u_1 \\ \vdots \\ \lambda u_n\end{bmatrix} + \begin{bmatrix}\lambda n_1 \\ \vdots \\ \lambda v_n\end{bmatrix} \\[1em]
&= \lambda\begin{bmatrix}u_1 \\ \vdots \\ u_n\end{bmatrix} + \lambda \begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix} \\[1em]
&= \lambda\vec{u} + \lambda\vec{v} &\blacksquare
\end{align}$$