You have a set with two things in it. That’s your basis. A linear combination allows you to add the two things together, and to λ scale them. That’s called spanning the basis.
The difference to a convex combination is:
- a convex combination lets you connect two points and “be” anywhere in between
- a linear combination actually opens up the whole plane that the two points lie in (a plane is a lot in 3-D but not much in higher D)
So I was listening to “When Doves Cry” and having some thoughts I could not have had without linear algebra:
- maybe I’m just like my father
- he’s worried he’s merely a convex combination of his parents—just 40% mom + 60% dad, or whatever
- are people doomed to that?
- maybe they could “slingshot” their way out of the trap by being “more mom than Mom” — a more general linear combination like 220% mom + −30% dad
- maybe when you meet other kids and teachers at school, you can take little pieces of them and make a new identity for yourself that way
- I think I did that, to an extent; copied 10% of my step-dad’s boorishness, 30% of my mom’s whimsy, −250% of everything about my step-mom; copied some teachers and friends too
- does that have something to do with creativity?
- like maybe your band can’t create a Truly New Sound, but it can choose a never-before-heard point inside the convex hull of bands that already existed.
- or maybe it can push outside the convex hull of music but sounds are mostly noise, ambient, or quiet out there.
- did Beefheart, Godspeed, and Cage reach out of the convex hull of existing music? From just a few dimensions outside to everything outside.
OK so a convex combination is probably not what Prince meant. But it was a related idea at least.
This flight of fancy was brought to me by: Linear Algebra Class. Mathematics is a totally different language than English. It’s more different from English than is Mandarin, Pormpuraaw, Tagalog, Aymara, Farsi, or Pirahã. That means you can think different thoughts once you learn mathematics. You can fathom what was unfathomable.Conceive what was inconceivable. See what was invisible. It also means that learning to “speak”this way sounds very strange.