A linear map from [math]$\mathbb{R}^2$[/math] to [math]$\mathbb{R}^2$[/math] is defined by the image of a basis. It transforms all the points of the plane, for example an image. By putting the coordinates of the images of the two image vectors in the starting basis, we obtain a table of 2x2 numbers. The linear application has a determinant, which is the ratio of the (algebraic) areas of the parallelograms of the image and the basis defined by the pairs of vectors. It also has a norm which is the largest norm of the image of a vector of norm 1. When its discriminant is positive, it also has two eigendirections, along which a vector and its image are aligned, their ratio being the eigenvalue. The trace is the sum of the eigenvalues, the determinant their product.

You can modify the images [math]$Me_1$[/math] and [math]$Me_2$[/math] of the basis vectors [math]$e_1$[/math] and [math]$e_2$[/math]. This modifies the associated matrix. You can move the point [math]$A$[/math] to see its image [math]$MA$[/math] and the sequence of images. A character is also available. You can observe the values ​​of the matrix, the values ​​of the determinant, the trace, the norm (associated with the shape of the ellipse, image of the unit circle). The discriminant and the eigenvalues ​​are also calculated when possible. Note the geometric sense of the proper directions: rotate A around the origin, when the matrix is ​​diagonalizable, [math]$A$[/math] "catches up" with [math]$MA$[/math], which is not the case if the discriminant is negative.