Standard basis vector ($e_i$)

The standard basis vector $e_i$ in ${\mathbb{F}}^n$ has coordinate one in position $i$ and zero elsewhere. Every vector $v=(v_1,\dots ,v_n)$ has the unique expansion $v={\sum }_{i=1}^n{v}_i{e}_i$.

For example, $e_2=(0,1,0{)}^{\top }$ in three dimensions. The outer product $e_i{e}_j^{\top }$ is the matrix unit with a single one in row $i$, column $j$, providing a standard basis for matrices as well. The word standard refers to the chosen coordinate representation; an abstract vector space does not come with a distinguished coordinate basis automatically.