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@jemoka / Jemoka Knowledge Base / raw/concept/kbhlength_of_basis_doesn_t_depend_on_basis.md
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--- title: "Length of Basis Doesn't Depend on Basis" source: https://www.jemoka.com/posts/kbhlength_of_basis_doesn_t_depend_on_basis/ --- 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 \(len(B_1)=len(B_{2})\). \(\blacksquare\)