Houjun Liu

Axler 1.B

Key Sequence

New Definitions

Results and Their Proofs

Questions for Jana

  • The way Axler presented the idea of “over” is a tad weird; is it really only scalar multiplication which hinders vector spaces without \(\mathbb{F}\)? In other words, do the sets that form vector spaces, apart from the \(\lambda\) used for scalar multiplication, need anything to do with the \(\mathbb{F}\) they are “over”? The name of the field and what its over do not have to be the same—“vector space \(\mathbb{C}^2\) over \(\{0,1\}\)” is a perfectly valid statement
  • If lists have finite length \(n\), then what are the elements of \(\mathbb{F}^{\infty}\) called? “we could think about \(\mathbb{F}^{\infty}\), but we aren’t gonna.”
  • Why is \(1v=v\) an axiom, whereas we say that some \(0\) exists? because we know 1 already, and you can follow the behavor of scalar multiplication
  • what’s that thing called again in proofs where you just steal the property of a constituent element?: inherits

Interesting Factoids

  • The simplest vector space is \(\{0\}\)