---
title: "Axler 2.C"
source: https://www.jemoka.com/posts/kbhaxler_2_c/
---

Key Sequence Because Length of Basis Doesn&rsquo;t Depend on Basis, we defined dimension as the same, shared length of basis in a vector space We shown that lists of the right length (i.e. dim that space) that is either spanning or linearly independent must be a basis&mdash;&ldquo;half is good enough&rdquo; theorems we also shown that \(dim(U_1+U_2) = dim(U_1)+dim(U_2) - dim(U_1 \cap U_2)\): dimension of sums New Definitions dimension Results and Their Proofs Length of Basis Doesn&rsquo;t Depend on Basis lists of right length are basis linearly independent list of length dim V are a basis of V spanning list of length of dim V are a basis of V dimension of sums Questions for Jana Example 2.41: why is it that \(\dim U \neq 4\)? We only know that \(\dim \mathcal{P}_{3}(\mathbb{R}) = 4\), and \(\dim U \leq 4\). Is it because \(U\) (i.e. basis of \(U\) doesn&rsquo;t span the polynomial) is strictly a subset of \(\mathcal{P}_{3}(\mathbb{R})\), so there must be some extension needed? because we know that \(U\) isn&rsquo;t all of \(\mathcal{P}_{3}\). Interesting Factoids
