[[
wikihub
]]
Search
⌘K
Explore
People
For Agents
Sign in
Explore
People
For Agents
Sign in
@jemoka / Jemoka Knowledge Base / wiki/concepts/length_of_basis_doesn_t_depend_on_basis.md
Suggest edit
Cancel
Submit suggestion
Title
Name
Note
--- 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’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 \(len(B_1)=len(B_{2})\). \(\blacksquare\)