## 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
- polynomial
- 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