3.2b Standard Unit Vector

Standard Unit Vector

The standard unit vectors for the vector-space ${\mathbb{R}}^n$ are a collection of vectors,
$\left\{\begin{array}{cccc}{\vec{e}}_1=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right],& {\vec{e}}_2=\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right],& \dots ,& {\vec{e}}_n=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right]\end{array}\right\},$
where each $i$ in ${\vec{e}}_i$ represents the index of the non-zero element.

Example

3D Space

In the 3D vector-space ${\mathbb{R}}^3$, the standard unit vectors are,
$\left\{\begin{array}{ccc}{\vec{e}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],& {\vec{e}}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],& {\vec{e}}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\end{array}\right\}.$