Linear algebra is typically explained using matrices. But matrix theory is just one possible perspective. Below, I describe an alternative approach to linear algebra.
Tadeusz Banachiewicz (1882–1954), a Polish astronomer living in Krakow, was passionate about calculating machines. From the 1920s, Banachiewicz developed a method for computations on tables of numbers, which was particularly easy to perform with arithmometers. In honor of Krakow, Banachiewicz named these computational objects cracovians. In the preface to Banachiewicz’s book, Cracovian Algebra (Rachunek krakowianowy, PWN 1959), it is noted that “some calculations, generally known to be very tedious, became a sheer entertainment for the executor.”
Similar to matrices, cracovians are represented as rectangular tables of numbers. The equality of cracovians, addition of cracovians, and multiplication of cracovians by a scalar are defined identically to their matrix counterparts. However, the multiplication of a cracovian by another cracovian is defined differently: the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian [thanks to andrewla from Hacker News for this definition]. In essence, each column of the left cracovian is multiplied by each column of the right cracovian.
Here’s an example (cracovians are enclosed in braces to distinguish them from matrices):
[I got the initial version of this post by translating my Polish text into English with Gemini. Gemini hallucinated the translation of this example. Now it should be OK. Thanks to everyone who noticed it :-)]
All such cracovians whose elements on the main diagonal are equal to 1 and other elements are zeros are denoted by the Greek letter τ (tau). Any such unit cracovian τ as the second factor of multiplication does not change the cracovian by which it is multiplied. As the first factor of multiplication, a unit cracovian τ transposes the cracovian by which it is multiplied:
For any such cracovians A and B that have the same number of rows, the identity holds:
Therefore, in the general case, cracovian multiplication is not commutative.
By convention, we agree that cracovian multiplication is left-associative, i.e.,
The product of cracovians has as many columns as the first factor, and as many rows as the last factor has columns.
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