https://youtu.be/yjBerM5jWsc?si=w0fzjCexdpGHVjQR #LinearAlgebra #MITOCW
#linearAlgebra
TIL "Linear Algebra Done Right" is now open access [1], and seems updated since the 2015 edition. I love that it gives proofs for the things that, in the words of Victor Klee, are obvious but also true.
https://library.oapen.org/bitstream/handle/20.500.12657/85067/978-3-031-41026-0.pdf
@typeswitch
tl;dr solution at bottom
I guess we need to clarify what the allowed operation is.
e.g. if I am allowed to whether any (natural?) linear combination of outcomes is probability 1/2, then I can just ask is 3*P(rolling i) = 1/2.
If the question I'm allowed to ask is
P(rolling i, j, or k) = 1/2, then we are essentially asking for an invertible 6x6 matrix with one row of 1s, and five rows containing a permutation of [1, 1, 1, 0, 0, 0]. In which case, again you are using 5 tests
That is certainly brute forceable by computer (at most (6 choose 3) choose 5 = 15,504 meaningfully different possibilities), but there may be a clever rank argument that it's possible or not.
Okay, I did some more thinking while doing my laundry, and here's a strategy:
If a + b + c = 0.5 and d + b + c = 0.5, then a = d. Repeat with modifications four more times to show that that all are equal to one another.
The problem with that is that doing it naively requires six tests, so seven equations, if we include that they all sum to one, my bad. So you need to take advantage of all the information you've gathered in the first four iterations and choose smartly in the last one.
Here is a solution written as a matrix equation
\(\begin{pmatrix}
1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 & 0 & 0\\
1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0\\
1 & 1 & 0 & 0 & 0 & 1 \\
0 & 1 & 1 & 1 & 0 & 0\\
\end{pmatrix}
\cdot \begin{pmatrix}
1/6 \\ 1/6 \\ 1/6 \\ 1/6 \\ 1/6 \\ 1/6
\end{pmatrix}
= \begin{pmatrix}
1 \\ 1/2 \\ 1/2 \\ 1/2 \\ 1/2 \\ 1/2
\end{pmatrix}
\)
One final comment, because I can't help myself. It shouldn't be too hard to generalise this pattern to other *even* sizes of dice.
#linearalgebra #probability
When I switched my major in college from physics to mathematics, I met with the undergraduate advisor for the department to sketch out courses, she (Kathy Davis at the University of Texas) said "you can never learn enough linear algebra."
As time has gone on, I keep going back to that as probably the deepest truth I've ever been told.
#mathematics #mathematicseducation #linearalgebra #universityoftexas
Linear Algebra and Optimization for Machine Learning (1 ed)
https://ebokify.com/linear-algebra-and-optimization-for-machine-learning-1-ed
The determinant of transvections (an update). New blog post on Freedom math dance.
In the previous post, I had explained how I could prove a general version of the classic fact that transvections have determinant 1. Here, I show than one can do more with less effort!
https://freedommathdance.blogspot.com/2025/11/the-determinant-of-transvections-update.html
Notes from the 2025 BLIS retreat.
https://www.cs.utexas.edu/~flame/BLISRetreat2025/Talks.html
#blis #hpc #supercomputing #appliedmath #linearalgebra #math #statistics
Ah, the age-old enigma: 🧙♂️ words that defy #translation. Clearly, the only solution is to whip out some 🧮 linear algebra! Because why struggle with #language when you can just slam it on a #matrix and call it a day? 🙄
https://aethermug.com/posts/linear-algebra-explains-why-some-words-are-effectively-untranslatable #challenges #linearalgebra #humor #HackerNews #ngated
Linear Algebra Explains Why Some Words Are Effectively Untranslatable
https://aethermug.com/posts/linear-algebra-explains-why-some-words-are-effectively-untranslatable
#HackerNews #LinearAlgebra #UntranslatableWords #LanguageMath #Linguistics #HackerNews
just in case anyone is interested in how to do complex linear algebra, including complex tensor products, without using complex numbers - using real linear operators \(J\) with \(J\circ J=-I\) instead of complex scalars - here are some links.
The TLDR version on a poster:
http://mwaa.math.indianapolis.iu.edu/Slides/coffman.pdf
Using a basis, matrices, and summing over indices - sections 4&5 of this paper:
https://users.pfw.edu/CoffmanA/pdf/basis1.pdf
Without a basis (but still a lot of notation) - Chapter 5 starting on page 195, with tensor products starting with Example 5.74 on page 208:
https://users.pfw.edu/CoffmanA/pdf/book.pdf
#LinearAlgebra #ComplexNumbers #NotASubToot
The determinant of transvections. — New blog post on Freedom Math Dance
A transvection in a K-vector space V is a linear map T(f,v) of the form x↦x+f(x)v, where f is a linear form and v is a vector such that f(v)=0. It is known that such a linear map is invertible, with inverse given by f and −v. More precisely, one has T(f,0)=id and T(f,v+w)=T(f,v)∘T(f,w). In finite dimension, these maps have determinant 1 and it is known that they generate the special linear group SL(V), the group of linear automorphisms of determinant 1.
When I started formalizing in Lean the theory of the special linear group, the question raised itself of the appropriate generality for such results. In particular, what happens when one replaces the field K with a ring R and the K-vector space V with an R-module?
https://freedommathdance.blogspot.com/2025/11/the-determinant-of-transvections.html
An Illustrated Introduction to Linear Algebra, Chapter 2: The Dot Product
https://www.ducktyped.org/p/linear-algebra-chapter-2-the-dot
#HackerNews #LinearAlgebra #DotProduct #IllustratedIntroduction #MathEducation #DataScience
Client Info
Server: https://mastodon.social
Version: 2025.07
Repository: https://github.com/cyevgeniy/lmst