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

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

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

$$\begin{align}
\vec{u} + \vec{v} + \vec{w} &=
\begin{bmatrix}u_1 \\ \vdots \\ u_n\end{bmatrix} + \begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix} + \begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}u_1 + v_1 \\ \vdots \\ u_n + v_n\end{bmatrix} + \begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}u_1 + v_1 + w_1 \\ \vdots \\ u_n + v_n + w_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}u_1 + (v_1 + w_1) \\ \vdots \\ u_n + (v_n + w_n)\end{bmatrix} \\[1em]
&= \begin{bmatrix}u_1 \\ \vdots \\ u_n\end{bmatrix} + \begin{bmatrix}v_1 + w_1 \\ \vdots \\ v_n + w_n\end{bmatrix} \\[1em]
&= \begin{bmatrix}u_1 \\ \vdots \\ u_n\end{bmatrix} + \left(\begin{bmatrix}v_1 \\ \vdots \\ v_n\end{bmatrix} + \begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix}\right) \\[1em]
&= \vec{u} + (\vec{v} + \vec{w}) &\blacksquare
\end{align}$$