Example 5.1.2.
The vectors \({\mathbf e}_1 = (1, 0)\) and \({\mathbf e}_2 = (0, 1)\) form a basis for \({\mathbb R}^2\text{.}\) Indeed, if \({\mathbf z} = (z_1, z_2)\text{,}\) then we can write
\begin{equation*}
{\mathbf z} = z_1 {\mathbf e}_1 + z_2 {\mathbf e}_2.
\end{equation*}
The vectors \({\mathbf e}_1\) and \({\mathbf e}_2\) are called the standard basis for \({\mathbb R}^2\text{.}\)