= Standard basis vector
{title2=$e_i$}
The <standard basis vector> $e_i$ in $\mathbb F^n$ has coordinate one in position $i$ and zero elsewhere. Every vector $v=(v_1,\ldots,v_n)$ 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.
Back to article page
