---
title: "Length of Basis Doesn't Depend on Basis"
type: concept
related: [Linear Independence, Finite Dimensional Vector Space, Basis]
source: https://www.jemoka.com/posts/kbhlength_of_basis_doesn_t_depend_on_basis/
confidence: high
status: active
---

Any two basis of finite-dimensional vector space have the same length.
constituents A finite-dimensional vector space \(V\) Basis \(B_1\), \(B_2\) be bases in \(V\) requirements Given \(B_1\), \(B_2\) are basis in \(V\), we know that they are both linearly independent and spans \(V\). We have that the length of linearly-independent list \(\leq\) length of spanning list.
Let&rsquo;s take first \(B_1\) as linearly independent and \(B_2\) as spanning:
We have then \(len(B_1) \leq len(B_2)\)
Swapping roles:
We have then \(len(B_2) \leq len(B_1)\)
As both of this conditions are true, we have that \(len(B_1)=len(B_{2})\). \(\blacksquare\)
