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?