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 $x_1 e_1 + \cdots + x_n e_n$.

Remarks

The coordinates of a vector in the standard basis are its entries. In $\R^3$, the standard basis vectors are often written $\mathbf{i}, \mathbf{j}, \mathbf{k}$. The column $j$ of an $m \times n$ matrix $A$ is $A e_j$. The standard basis of $\R^n$ is an orthonormal basis for the dot product.

Explanations

L1 human

The standard basis of $F^n$ consists of the vectors with one entry $1$ and all other entries $0$. In the plane, these are the unit arrows along the $x$-axis and the $y$-axis. The coordinates of a vector in this basis are just its entries.