The hottest Linear Algebra Substack posts right now

And their main takeaways
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Top Science Topics

β€œLights Out” on a triangular grid

A Piece of the Pi: mathematics explained β€’ 30 implied HN points β€’ 22 Mar 26
πŸ”¬ Science Linear Algebra
  1. The triangular Lights Out game reduces to linear algebra over the field with two elements: pressing a button toggles bits mod 2, pressing a button twice cancels, order doesn’t matter, and any solution is a subset of buttons pressed once.
  2. Solvability and uniqueness depend on the kernel of the toggle map: if the kernel is only the empty set (β„“=0) then every starting state has a unique solution, which occurs for certain side lengths such as 1, 3, 4, 7, 8, 9, 11, 15, 16, 17, 20, and 21.
  3. If the kernel is nontrivial (β„“>0) there are nonzero button patterns that have no effect and some starting configurations cannot be solved; the kernel is a 2^β„“-sized vector space over GF(2) and its patterns often form visually striking shapes like the SierpiΕ„ski triangle.

Domain Transformations: The Art of Finding Easier Spaces

Software Bits Newsletter β€’ 103 implied HN points β€’ 05 Jan 26
πŸ•Ή Technology Linear Algebra
  1. Transform hard problems into easier ones by moving to a different domain, doing the simpler computation there, and (if needed) transforming the result back; this is worth it when the transform cost plus the easier computation is less than solving the original problem.
  2. Use well-known transforms to fix numerical and computational issues: log-space turns tiny-product underflow into stable sums (use the log-sum-exp trick to add probabilities safely), Fourier turns convolution into cheap pointwise multiplication, and embeddings or kernels lift data so linear methods work.
  3. Always check that a transform preserves what you need and that the round-trip cost is justified; the best algorithms exploit problem structure by finding the space where the computation becomes simple.

Linearity: Why Batching Works

Software Bits Newsletter β€’ 103 implied HN points β€’ 03 Jan 26
πŸ•Ή Technology Linear Algebra
  1. Linearity lets you process many inputs as one big matrix multiply, so batching is nearly free and GPUs can run large batches with high efficiency.
  2. Differentiation is linear, so per-sample gradients can be summed and scaled β€” enabling gradient accumulation, distributed training, and efficient backprop.
  3. Non-linearities are required for expressivity, so networks interleave cheap, element-wise nonlinear functions with batch-friendly linear layers and prefer operations (like LayerNorm) that preserve batching advantages.

Why is the Golden Ratio Hiding in the Fibonacci Sequence?

The Palindrome β€’ 4 implied HN points β€’ 30 Jan 26
🚌 Education Linear Algebra
  1. The Fibonacci sequence has a simple closed-form (Binet's) formula that uses the golden ratio phi (Ο†) and its conjugate psi (ψ) to compute Fn directly. It yields exact integers even though Ο† and ψ are irrational.
  2. Generating functions turn the recurrence into the rational function F(x)=x/(1-x-x^2), and partial fraction decomposition expresses F(x) as a sum of two geometric series, which leads to Binet's formula.
  3. The recurrence can also be encoded by a 2Γ—2 Fibonacci matrix whose eigenvalues are Ο† and ψ; diagonalizing that matrix and raising it to the nth power gives the same closed-form result, and computing matrix powers is often numerically more stable than directly evaluating Binet's formula.

The maximum number of SETs for 12 cards

A Piece of the Pi: mathematics explained β€’ 90 implied HN points β€’ 10 Feb 25
πŸ”¬ Science Linear Algebra
  1. The game SET uses 81 cards that have four qualities: quantity, shading, color, and shape. Players look for sets of three cards where each quality is either all the same or all different.
  2. SET can be understood through linear algebra, where each card is represented as a four-dimensional vector. If the vectors for three cards add up to zero, they form a valid set.
  3. Recent research showed that with 12 cards, a maximum of 14 sets can be formed, and they provided proofs for similar results with fewer cards. This reveals interesting mathematical properties of the game.
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A Linear Algebra Trick for Computing Fibonacci Numbers Fast

Confessions of a Code Addict β€’ 158 HN points β€’ 05 Nov 23
πŸ”¬ Science Linear Algebra
  1. A linear algebra technique can be applied to compute Fibonacci numbers quickly with a logarithmic time complexity.
  2. Efficient algorithms like repeated squaring can compute powers of matrices in logarithmic time, improving performance for Fibonacci number calculations.
  3. A closed form expression using the golden ratio offers a direct method to compute Fibonacci numbers, showing different approaches with varied performance.

Explaining "Taking features out of superposition with sparse autoencoders"

From AI to ZI β€’ 19 implied HN points β€’ 16 Jun 23
πŸ”¬ Science Linear Algebra
  1. Explanations of complex AI processes can be simplified by using sparse autoencoders to reveal individual features.
  2. Sparse and positive feature activations can help in interpreting neural networks' internal representations.
  3. Sparse autoencoders can be effective in reconstructing feature matrices, but finding the right hyperparameters is important for successful outcomes.

The Camel Principle: Why Adding Zero is the Most Powerful Trick in Mathematics

The Palindrome β€’ 1 implied HN point β€’ 12 Jan 26
🚌 Education Linear Algebra
  1. The camel principle is the idea that you can add zero in clever ways to transform problems, and that tiny trick can unlock big simplifications.
  2. Adding zero is essential because it helps rewrite expressions, simplify derivations, and connect different methods across mathematics and machine learning.
  3. A practical workshop can teach these foundations by building linear regression from scratch, covering vectors, vectorized code, optimization, and gradient descent with notebooks and recordings for practice.

The Role of Advanced Mathematics in AGI: Could Abstract Theories Pave the Way?

Deep-Tech Newsletter β€’ 1 HN point β€’ 12 Jul 24
πŸ”¬ Science Linear Algebra
  1. The AI industry is investing heavily in large language models and AGI, but faces financial challenges and uncertainty in meeting high expectations.
  2. To achieve AGI, more advanced mathematical techniques beyond current ML algorithms like gradient descent may be needed, with Category Theory showing promise.
  3. Barriers exist in understanding Category Theory for AGI due to its abstract nature, but efforts are being made to empower AI researchers and engineers with necessary mathematical knowledge.

Epsilons, no. 3: The LU decomposition

The Palindrome β€’ 3 implied HN points β€’ 27 Mar 23
πŸ”¬ Science Linear Algebra
  1. Matrix factorizations are a key part of linear algebra, used for inverting matrices and simplifying determinants.
  2. The LU decomposition method involves breaking a matrix into upper and lower triangular forms.
  3. Linear algebra helps in solving systems of linear equations by transforming them into echelon form using operations like multiplying by scalars and adding equations.