#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof

#show: ams-article.with(
  title: [Week 2],
  authors: (
    (
      name: "Youwen Wu",
      organization: [University of California, Santa Barbara],
      email: "youwen@ucsb.edu",
      url: "https://youwen.dev",
    ),
  ),
  bibliography: bibliography("refs.bib"),
)

= Vectors, linear combinations, spans, matrix-vector product.

- Consider a whole new way of looking at linear systems
- Add vectors entrywise, head to tail
- Multiply vectors via scaling
- A more flexible way to draw a line. For a line through point $p$, in direction $arrow(d)$, use $arrow(p) + t arrow(d), t in RR$. Intuition: Add a vector $arrow(p)$ pointing to point $p$ and compose a vector pointing in the intended direction $arrow(d)$ head to tail.

A linear combination is

$ arrow(y) = sum_(k=1)^n alpha_n arrow(v)_n $