by Jean Barbet | May 23, 2021 | Algebra, Geometry
The linear transformations of the Euclidean plane are the invertible linear applications, i.e. of non-zero determinant. They allow us to move from one basis of the plane to another, and the orthogonal transformations, i.e. the vectorial isometries, exchange the...
by Jean Barbet | May 8, 2021 | Algebra, Geometry, Non classé
The representation of the Euclidean plane as the Cartesian product \(\mathbb R^2\) allows us to decompose any vector of the plane into two coordinates, its abscissa and its ordinate. This decomposition is linked to a particular and natural “representation...
by Jean Barbet | Mar 25, 2021 | Algebra, Geometry, Non classé, Number Theory
Descartes’ analytical method, which allows the Euclidean plane to be represented as the Cartesian product $ \mathbb{R}^2 $ through the theory of real numbers, also makes it possible to represent Euclidean space as the Cartesian product $ \mathbb{R}^3 =...
by Jean Barbet | Mar 20, 2021 | Algebra, Geometry, Non classé, Number Theory
The complex multiplication naturally extends to a multiplication in four dimensions, which defines on the space $ \mathbb{R}^4 $ the structure of the algebra $ \mathbb{H} $ of Hamilton’s quaternions. This multiplication can be interpreted geometrically using the...
by Jean Barbet | Mar 12, 2021 | Algebra, Non classé, Number Theory
Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular, we can determine which are the usual prime numbers that...
by Jean Barbet | Feb 13, 2021 | Algebra, Geometry, Non classé
Introduction In Vector angles: geometric intuition and algebraic definition, we defined and described the group of Euclidean plane vector angles algebraically, using an equivalence relation on unit vectors. Just as we can measure lengths, we learn at primary school...
