Objectives and competences

To understand the mathematical and algebraic concepts that underpin matrix algebra, linear transformations, determinant theory, and systems of linear equations. To solve problems in order to develop students' reasoning skills. At the end of the course unit, students are expected to be able to:
-  solve any system of linear equations with any number of unknowns and equations;
-  perform operations with linear transformations.

Teaching Methodologies

Lectures focused on structured presentation of concepts. Problem-solving sessions to consolidate learning.
Active learning strategies to promote autonomy and critical thinking.

Syllabus

Matrix Algebra
Definition. Type and order of a matrix. Matrix addition and scalar multiplication. Matrix multiplication. Transpose of a matrix.
Inverse matrix and Gaussian elimination method. Properties of the inverse matrix. Matrix: diagonal, scalar, triangular, symmetric, hemisymmetric.

Linear Transformations (LT)
Definition. Kernel and image of a LT. Basis of a linear space and independence between vectors. Matrix of an LT.
Change of basis. Transition matrix.

Theory of Determinants (Det) Definition. Sarrus' rule.
Laplace's theorem. Properties of Det.
Adjoint matrix of a matrix. Calculation of the inverse matrix from the adjoint.

Systems of linear equations (SLE) Definition.
Rouché's theorem to ascertain the possibility of a SLE.
Kronecker's theorem for determining the matrix characteristic of the system. Cramer's system. Homogeneous system.
Solving SLEs based on the matrix condensation technique: Gauss-Jordan method.
Solving SLEs based on Cramer's rule. Discussion and classification of SLEs as a function of the real values of some parameters.