Exercise 5.1ΒΆ
We need to prove the associativity law for \((\mathbb{R}^2, +)\); that is, we need to show that \(\forall \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in \mathbb{R}^2.\; (\boldsymbol{x} + \boldsymbol{y}) + \boldsymbol{z} = \boldsymbol{x} + (\boldsymbol{y} + \boldsymbol{z})\).
This result follows naturally from associativity of addition in \(\mathbb{R}\):
\[\begin{split}(
\begin{bmatrix}x_1\\x_2\end{bmatrix} +
\begin{bmatrix}y_1\\y_2\end{bmatrix}
) +
\begin{bmatrix}z_1\\z_2\end{bmatrix}
&=
\begin{bmatrix}x_1 + y_1\\x_1 + y_2\end{bmatrix} +
\begin{bmatrix}z_1\\z_2\end{bmatrix}
\\ &=
\begin{bmatrix}x_1 + y_1 + z_1\\x_1 + y_2 + z_2\end{bmatrix}
\\ &=
\begin{bmatrix}x_1\\x_2\end{bmatrix} +
\begin{bmatrix}y_1 + z_1\\y_1 + z_2\end{bmatrix}
\\ &=
\begin{bmatrix}x_1\\x_2\end{bmatrix} +
(
\begin{bmatrix}y_1\\y_2\end{bmatrix} +
\begin{bmatrix}z_1\\z_2\end{bmatrix}
)\end{split}\]