Exercise 5.2

We need to come up with a recipe for finding the inverse of a vector in \((\mathbb{R}^2, +)\): that is, given a vector, find another vector such that the sum of these two vectors is the the zero vector.

Suppose we have an \(\boldsymbol{x} \in \mathbb{R}^2\), so \(\boldsymbol{x} = (x_1, x_2)\). If we add it to some other vector \(\boldsymbol{y} = (y_1, y_2)\), we get \((x_1 + y_1, x_2 + y_2)\). For this sum to be equal to the zero vector we have to choose \(y_1, y_2\) so that the following two equations are satisfied:

\[ \begin{align}\begin{aligned}x_1 + y_1 = 0\\x_2 + y_2 = 0\end{aligned}\end{align} \]

The solution is therefore

\[ \begin{align}\begin{aligned}y_1 = -x_1\\y_2 = -x_2\end{aligned}\end{align} \]

or simply \(\boldsymbol{y} = -\boldsymbol{x}\). That is, you can invert a vector in \((\mathbb{R}^2, +)\) by performing a scalar multiplication by \(-1\).