Algebra Lesson 2 Linear Applications Endomorphisms and Diagonalization

After exploring the structure of vector spaces, their subspaces, bases, and dimensions, the next natural step is to study how these spaces are transformed. Topic 2 focuses on linear maps, which are the rules that allow vectors to be transferred from one space to another while preserving their algebraic structure.

A linear map is not just a function: it is a tool that preserves addition and scalar multiplication, and can be represented by matrices. Through them, we introduce fundamental concepts such as the kernel (the set of vectors that are mapped to zero) and the image (the set of reachable vectors), which allow us to classify maps as injective, surjective, or bijective.

When the domain and codomain coincide, we speak of endomorphisms, and if they are also invertible, of automorphisms. These cases are especially important because they allow us to study the internal structure of a vector space through its eigenvalues and eigenvectors, which reveal invariant directions under the transformation.

The diagonalization of an endomorphism is one of the most powerful processes in linear algebra: it allows us to simplify the matrix that represents it, facilitating calculations and analysis. This technique is directly connected to bilinear and quadratic forms, which appear in geometry, physics, and optimization. Through them, we can classify curves and surfaces such as conics (circles, ellipses, parabolas, hyperbolas) and quadrics (ellipsoids, hyperboloids, paraboloids), according to the eigenvalues of the associated tensors.