Audio note: this article contains 98 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Why are linear functions between finite-dimensional vector spaces representable by matrices? And why does matrix multiplication compose the corresponding linear maps? There's geometric intuition for this, e.g. presented by 3Blue1Brown. I will alternatively present a category-theoretic analysis. The short version is that, in the category of vector spaces and linear maps, products are also coproducts (hence biproducts); and in categories with biproducts, maps between biproducts factor as (generalized) matrices. These generalized matrices align with traditional numeric matrices and matrix multiplication in the category of vector spaces. The category-theoretic lens reveals matrices as an elegant abstraction, contra The New Yorker.

I'll use a standard notion of a vector space over the field <span>_mathbb{R}_</span>. A vector space has addition, zero, and scalar multiplication defined, which have the standard commutativity/associativity/distributivity properties. The category <span>_mathsf{Vect}_</span> has as objects vector spaces (over the field <span>_mathbb{R}_</span>), and as morphisms linear maps. A linear map <span>_f : U rightarrow V_</span> between vector spaces U, V satisfies <span>_f(u_1 + u_2) = f(u_1) + f(u_2)_</span> and <span>_f(au) [...]

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First published:
November 15th, 2025

Source:
https://www.lesswrong.com/posts/7wstHFRn3bHzSN2z3/matrices-map-between-biproducts

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Narrated by TYPE III AUDIO.

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Images from the article:

Diagram showing mappings between vector spaces with functions and projections.

Mathematical diagram showing tensor product decomposition with mappings f, g, and projections π₁, π₂.

Diagram showing mathematical representations of sets U, V, and W with mappings.

Matrix notation showing elements h with subscripts i,j arranged in n by m format.

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“Matrices map between biproducts” by jessicata

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