## 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

## Results and Their Proofs

- Length of Basis Doesn’t Depend on Basis
- lists of right length are basis
- 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~~because we know that \(U\) isn’t all of \(\mathcal{P}_{3}\).*some*extension needed?