## Key Sequence

- we defined basis of a vector space—a linearly independent spanning list of that vector space—and shown that to be a basis one has to be able to write a write an unique spanning list
- we show that you can chop a spanning list of a space down to a basis or build a linearly independent list up to a basis
- because of this, you can make a spanning list of finite-dimensional vector spaces and chop it down to a basis: so every finite-dimensional vector space has a basis
- lastly, we can use the fact that you can grow list to basis to show that every subspace of \(V\) is a part of a direct sum equaling to \(V\)

## New Definitions

I mean its a chapter on bases not sure what you are expecting.

## Results and Their Proofs

- a list is a basis if you can write every memeber of their span uniquely
- every finite-dimensional vector space has a basis
- dualing basis constructions
- every subspace of \(V\) is a part of a direct sum equaling to \(V\)

## Questions for Jana

- Is the subspace direct sum proof a unique relationship? That is, is every complement \(W\) for each \(U \subset V\) unique?