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’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 \(len(B_1)=len(B_{2})\). \(\blacksquare\)