Exercise 5.3

Let \(\boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^2, k \in \mathbb{R}\). We will write \(x_1\) for the first component of \(\boldsymbol{x}\), \(x_2\) for the second component of \(\boldsymbol{x}\), and so on.

Then:

\[\begin{split}k (\boldsymbol{x} + \boldsymbol{y}) &= k (\begin{bmatrix}x_1\\x_2\end{bmatrix} + \begin{bmatrix}y_1\\y_2\end{bmatrix}) \\ &= k (\begin{bmatrix}x_1 + y_1\\x_2 + y_2\end{bmatrix}) \\ &= \begin{bmatrix}k(x_1 + y_1)\\k(x_2 + y_2)\end{bmatrix} \\ &= \begin{bmatrix}kx_1 + ky_1\\kx_2 + ky_2\end{bmatrix} \\ &= \begin{bmatrix}kx_1\\kx_2\end{bmatrix} + \begin{bmatrix}ky_1\\ky_2\end{bmatrix} \\ &= k \begin{bmatrix}x_1\\x_2\end{bmatrix} + k \begin{bmatrix}y_1\\y_2\end{bmatrix} \\ &= k\boldsymbol{x} + k\boldsymbol{y}\end{split}\]