Computational Linear Algebra

This repository contains materials related to the Practical an Theoretical classes from Computational Linear Algebra course. Below is a summary of the topics covered in the program:

Content

Vector Spaces and Linear Transformations

• Definition of real vector spaces.
• Subspaces, generating systems, and linear independence.
• Bases and dimension of a vector space.
• Linear transformations and their matrix representation.
• Kernel, image, co-kernel, and co-image of a linear transformation.

Norms and Linear Systems

• Vector and matrix norms.
• Cauchy-Schwarz inequality and triangular inequality.
• Error and conditioning of matrices.
• Solution of linear systems through Gaussian elimination and LU factorization.
• Orthogonal matrices and QR factorization.

Eigenvalues and Eigenvectors

• Basic properties of eigenvalues and eigenvectors.
• Gerschgorin's theorem.
• Diagonalization of matrices and eigenvector bases.
• Eigenvalues of symmetric matrices and the spectral theorem.
• Numerical methods for eigenvalue calculation (power method, QR algorithm).

Iterative Methods and Positive Definite Matrices

• Iterative methods for linear systems (Jacobi, Gauss-Seidel, SOR).
• Krylov subspace and conjugate gradient method.
• Positive definite matrices and Cholesky factorization.

Matrix Decompositions and Applications

• Singular value decomposition (SVD).
• Generalized inverse and Schur decomposition.
• Jordan canonical form.
• Bilinear forms and inner products.
• Least squares problems and approximation and interpolation.

This repository provides additional resources, including example codes, reading materials, and practical exercises, to complement the study of these topics.