Standard basis

Description

Let $F$ be a field and let $n \ge 1$ be an integer. For $i = 1, \dots, n$, let $e_i \in F^n$ be the vector with entry $1$ in position $i$ and $0$ in the other positions. The standard basis of the coordinate space $F^n$ is the family $(e_1, \dots, e_n)$. It is a basis of $F^n$, and each $x = (x_1, \dots, x_n)$ is equal to $x_1 e_1 + \cdots + x_n e_n$.