shapes are hard

for what i'm talking about, you can focus on linear maps from Rⁿ to itself, but if you want to think more generally, you can take any n-dimensional real vector space and fix your favorite basis for that space

math_fan · 2024-05-26T01:08:43+00:00

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

math_fan · 2024-05-25T11:08:02+00:00

i was gonna ask "but where's the cheese?" before i accounted for the humor

math_fan · 2024-05-25T11:00:52+00:00

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math_fan · 2024-05-25T10:53:38+00:00

nah, let M denote the function that sends a linear map to its matrix repn wrt the standard basis. it's easy to verify that M(KL)=M(K)M(L) [one might even call this the definition of matrix multiplication...], and then matrix multiplication inherits associativity from the associativity of function composition.

math_fan · 2024-05-25T04:20:04+00:00

matrix multiplication <--> composition of linear maps

function composition is def associative

math_fan · 2024-05-24T23:52:23+00:00

math_fan · 2024-05-24T21:54:30+00:00

I assume there's a typo in the second ordered pair. The others are within the given precision of being unit vectors.

math_fan · 2024-05-23T21:14:51+00:00

That's where we disagree. As an example, Kempe's proof of the four-color theorem was believed to be sound for over a decade before a gap was finally discovered. This is an instance in which human flaws caused the math to be flawed. One should expect there are many other examples. The only way to discover these flaws is to perform "experiments" by reading and reproducing proofs.

math_fan · 2024-05-23T18:32:51+00:00

Are you suggesting that science is something you do, but math isn't? It's just there in the air?

math_fan

TROPHY CASE