---
date: '2024-12-10'
description: also known as linear space, a set of vectors.
id: Vector space
modified: 2026-06-05 15:08:21 GMT-04:00
tags:
  - math
title: Vector space
created: '2024-12-10'
published: '2024-12-10'
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slug: thoughts/Vector-space
permalink: https://aarnphm.xyz/thoughts/Vector-space.md
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---
Not to be confused with [[thoughts/Vector calculus#vector field]], but a vector space is a set whose vectors can be scaled by any given _scalars_

A vector space have all basic properties of a set, including associative, commutative, identity, inverse, distributivity

## coordinates, and subspace

We can then take a look for a set of $G$ of a $F$-vector space $V$, the following properties:

### linear combination

$a_{1} \mathbf{g}_1 + a_{2} \mathbf{g}_2 + \cdots + a_{n} \mathbf{g}_n$

where scalars $a_1, \ldots a_{n}$ is the _coefficient_ of the linear combination

### linear independent, span, basis, subspace

See also [[thoughts/university/twenty-four-twenty-five/sfwr-4ml3/tut/tut1|linearly dependent, span and basis]]