Axler 2.C

Key Sequence Because Length of Basis Doesn’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—“half is good enough” 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’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’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’t all of \mathcal{P}_{3}. Interesting Factoids