Properties of scalar multiplication

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