---
tag: 059THXBRBB
kind: statement
role: proposition
name: Isomorphism with the coordinate space
msc: [15A03]
uses: [0BMVBHTWV2, 0XSJ25Z9M4]
sources:
  - url: https://en.wikipedia.org/wiki/Coordinate_vector
    locator: The standard representation
---

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$.
Then the map
$$V \to F^n, \qquad v \mapsto [v]_B,$$
that sends each vector to its [[coordinate vector|04CYQHN6MC]] in $B$, is a [[linear isomorphism|0B8T10CE7N]].
Its inverse sends $(a_1, \dots, a_n)$ to $a_1 b_1 + \cdots + a_n b_n$.

## Remarks

The isomorphism depends on the choice of the basis.
It sends $b_i$ to the standard basis vector $e_i$.
Through this map, each statement about $F^n$ gives a statement about $V$.

## Explanations

### L1 human

Choosing a basis turns every vector into a list of $n$ numbers.
This translation respects addition and scaling, and it loses no information.
So each $n$-dimensional space is a copy of $F^n$.