Axler 2.B

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

basis and criteria for basis

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
  • all spanning lists contains a basis of which you are spanning
  • a linearly independent list expends to a basis
• 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?