Axler 2.A

Key Sequence

• we defined the combination of a list of vectors as a linear combination and defined set of all linear combination of vectors to be called a span
• we defined the idea of a finite-dimensional vector space vis a vi spanning
• we took a god-forsaken divergence into polynomials that will surely not come back and bite us in chapter 4
• we defined linear independence + linear dependence and, from those definition, proved the actual usecase of these concepts which is the Linear Dependence Lemma
• we apply the Linear Dependence Lemma to show that length of linearly-independent list \(\leq\) length of spanning list as well as that finite-dimensional vector spaces make finite subspaces. Both of these proofs work by making linearly independent lists—the former by taking a spanning list and making it smaller and smaller, and the latter by taking a linearly independent list and making it bigger and bigger

New Definitions

• linear combination
• span + “spans”
• finite-dimensional vector space
  • infinite-demensional vector space
  • finite-dimensional subspaces
• polynomial
  • \(\mathcal{P}(\mathbb{F})\)
  • \(\mathcal{P}_{m}(\mathbb{F})\)
  • degree of a polynomial \(\deg p\)
• linear independence and linear dependence
• Linear Dependence Lemma

Results and Their Proofs

• span is the smallest subspace containing all vectors in the list
• \(\mathcal{P}(\mathbb{F})\) is a vector space over \(\mathbb{F}\)
• the world famous Linear Dependence Lemma and its fun issue
• length of linearly-independent list \(\leq\) length of spanning list
• subspaces of inite-dimensional vector spaces is finite dimensional

Questions for Jana

• obviously polynomials are non-linear structures; under what conditions make them nice to work with in linear algebra?
• what is the “obvious way” to change Linear Dependence Lemma’s part \(b\) to make \(v_1=0\) work?
• for the finite-dimensional subspaces proof, though we know that the process terminates, how do we know that it terminates at a spanning list of \(U\) and not just a linearly independent list in \(U\)?
• direct sum and linear independence related; how exactly?

Interesting Factoids

I just ate an entire Chinese new-year worth of food while typing this up. That’s worth something right