Geometry 1

The course aims to provide students with some knowledge and skills in linear algebra. Starting from the notion of finite dimensional vector space on any field, we arrive at solving the systems of linear equations with the Gauss-Jordan method. Another goal is to study linear and bilinear applications, illustrating the notion of a representative matrix and the related problems of diagonalization. The bilinear applications are used to investigate Euclidean vector spaces (real and complex) and self-adjoint operators, for which the spectral theorem is fully proved.

At the end of the course, students will have acquired the following skills:
1. they will be able to solve systems of linear equations;
2. they will be able to apply the theory of finite dimensional vector spaces, recognizing vector subspaces and determining their bases;
3. they will be able to study linear applications, determining the representative matrix, the kernel and the image;
4. they will be able to apply some aspects of the theory of diagonalization of endomorphisms and matrices, based on the search for eigenvalues and eigenvectors;
5. they will know how to work in spaces with a positive definite inner product (also called Euclidean spaces) and apply elementary notions of Euclidean geometry;
6. they will know how to recognize self-adjoint operators and will be able to diagonalize them, determining an orthonormal basis of eigenvectors by means of the spectral theorem (real and complex).