---
tag: 0V7ZD6P7RK
kind: object
name: Standard basis
aliases: [canonical-basis]
msc: [15A03]
instance_of: [0KCQ9Y3QKA]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_basis
    locator: lead section
same_as:
  wikidata: Q2000912
  wikipedia: 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|0KCQ9Y3QKA]] 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.