Math for everyone
Mathematical content accessible to everyone
Polynomials... viewed as vectors
What could a quadratic polynomial possibly have in common with a vector in three-dimensional space? At first glance, nothing: they are different kinds of objects. Yet both have the same form—each is described by a triple of numbers. Better still, calculations with one correspond to calculations with the other!
Drawing a spider's web
Have you ever found yourself doodling spiderwebs in the margin of a calculation that was going nowhere? Not like an entomologist (since spiders are not insects), but however you please, subject to a few simple geometric rules...
Words are vectors!
Comparing two vectors is rather like comparing two texts. This analogy proves highly relevant to the study of a literary corpus: combined with computing power, the tools of linear algebra can be used to compare two texts or measure their similarity.
Composing geometric transformations
Some geometric problems, however complicated they may look, quickly become clear once geometric transformations are brought in, and are often solved by composing them. To do this, it helps to recast them in vector terms.
Collinearity, coplanarity, concurrency... it's all the same story!
Points and lines in the plane are dual notions: theorems about collinear points correspond to theorems about concurrent lines. This duality can be defined geometrically. It even extends to space, through coplanarity.
Geometry without figures
Michel Chasles dreamed of it; the theory of vector spaces now makes it possible: we can do geometry without drawing a single figure. Geometric and algebraic viewpoints thus coexist, and everyone can choose whichever feels most comfortable!
From vector spaces to affine spaces… and back again!
Linear algebra arose from the need to provide a framework for ordinary geometry. From a computational standpoint, it has been a success! By making certain operations and manipulations simpler and more systematic, it streamlines geometric reasoning and makes it more rigorous.
Linear maps: the "hard core" of linear… algebra
Linear maps are an essential concept in the theory of vector spaces. They have many uses, including designing and solving games.
“The” dimension: not such an obvious idea!
The notion of dimension can be glimpsed in Euclid, then takes clearer shape with Descartes before branching out according to the subject at hand: analytic geometry, vector spaces or topology. There is a whole host of “dimensions”! Here, the focus is on the dimension of vector spaces, due to Georg Hamel.
Yannis Xenakis's "musical vector spaces" | Tangente
In 1963, the celebrated composer Yannis Xenakis introduced a kind of arithmetic for musical composition, based on the notion of mathematical structure—in particular, that of a vector space. This artistic contribution lends itself wonderfully to scientific analysis.
Matrix algebra in our images – Maths brief | Tangente
Matrices and vectors have become indispensable mathematical and computational tools for digital image processing.
The vector art manifesto – Maths brief | Tangente
The Scowcza collective creates works of vector art. You have already come across several of them in the articles in this issue
Gram–Schmidt method – Math brief | Tangente
Discover the celebrated Gram–Schmidt method for constructing orthonormal bases of vector spaces
Cross product and scalar triple product
Defined on three-dimensional Euclidean space, these two products grew out of Hamilton's quaternions. They have become useful tools in geometry and mechanics.
The slow emergence of Euclidean spaces
The axiomatic definition of the dot product provided a rigorous and fruitful framework for studying metric properties—those involving distance, angle, and orthogonality. This concept did not emerge clearly until the 1920s.
Translations and rotations – Maths brief | Tangente
Translations and rotations have many applications—and not only in geometry!
First examples of vector spaces (2)
Every good vector space E needs a base field K. But what exactly is a field?
Gaston Darboux: geometer and artist
Gaston Darboux, whom we commemorate this year, died a century ago. Throughout his life, he remained "a young man of the rarest learning and the highest promise," and he left behind a considerable body of work. His geometric results on orthogonal surfaces are spectacular.
Cryptographic machines: a history of encryption
Since antiquity, people have sought to ensure the secrecy of their correspondence through encryption. Because simple methods were easily decrypted, more sophisticated ones were invented, but they were difficult to use by hand. Machines therefore became essential.
The fascination of gears: geometry and mechanics
Gears are both objects of fascination and symbols of mechanics. Determining the shape of their teeth is a sophisticated mathematical problem. Yet this long history, which stretches back to antiquity and is still being written, remains remarkably little known!