Álgebra Linear e Geometria Analítica

Knowledge of the subject of Mathematics in secondary education.

The curricular unit has theoretical and theoretical-practical classes. Theoretical classes take place in an essentially expository way, approaching the themes foreseen in the program, prevailing a strong interaction between the concepts and their concrete application. The theoretical-practical classes will be aimed at solving problems and practical cases under the guidance of the teacher. The exercises will be performed individually or in small groups. The teaching of the curricular unit is complemented by student service periods.

The teaching of Mathematics in general should facilitate mathematical communication, reflective thinking, the application of mathematical techniques to problem solving, critical analysis of the results obtained, and finally, interdisciplinarity. One of the teaching objectives of the 1st year, Linear Algebra and Analytic Geometry discipline is to provide the basic foundations of mathematical methods, usually applied in the areas of Engineering, used by the various disciplines of the Degree in Biomedical Engineering.

It is intended that students develop abilities (skills) of algebraic manipulation and independent and analytical reasoning and the ability to apply mathematical concepts in solving practical problems.

1. Complex Numbers (revision)

2. Systems of Linear Equations and Matrices

• Application of the study of linear equations systems to solving problems linked to Biomedical Engineering.
• Matrix concept.
• Special matrices (line matrix, column matrix, triangular matrix, diagonal matrix, scalar matrix, identity matrix, transposed matrix and conjugate matrix).
• Operations with matrices and some properties.
• Invertible matrices.
• Matrix notation of  linear equations system.
• Characteristic of a matrix. Characteristic calculation using the Gaussian elimination method.
• Resolution of linear systems using the Gaussian elimination method.
• Systems with parameters. Discussion of systems with parameters.
• Inverse matrix. Calculation of the inverse matrix by the Gauss-Jordan method.

3. Determinants

• Introduction of determinant.
• Second order determinant: definition and properties.
• Third order determinant: definition using first-line development; development along one of the lines – “matrix of signals”. Sarrus Rule.
• Determinant of order n; definition.
• Minors, complementary minors, and algebraic complements. Laplace’s theorem and its generalization. Characteristics of a matrix and order of minors.
• Calculation of the determinant of a matrix transforming it into a triangular matrix using the Gaussian elimination method.
• Applications of determinants: adjunct matrix and inverse matrix; systems of linear equations. Cramer’s Rule.

4. Eigenvalues and eigenvectors

• Introduction of the concept of eigenvalue and eigenvector of a linear application.
• Definition of eigenvalue and eigenvector. Finding eigenvalues and eigenvectors.
• Diagonal matrix representation of a linear application. Diagonalizing matrix and diagonalizing matrix concept.
• Characteristic polynomial and characteristic equation of a matrix. Cayley-Hamilton theorem and some applications

 5. Analytical Geometry

• Dot and cross product between vectors.
• Equations of lines and planes.
• Intersection of lines and planes.
• Relative position between lines and planes.
• Angles between geometric identities.

The student can opt for a distributed assessment or a final exam assessment.

Distributed assessment (2 test + work):

• The first test will assess the following content - Complex numbers, systems of linear equations and matrices and determinants. You will be awarded a grade of 8 points, with no minimums, to be taken in week 9 or 10 during TP classes.
• The second test will evaluate the following contents – Determinants and eigenvalues ​​and eigenvectors. It will have a quotation of 8 values, with a minimum of 3 values, to be carried out in normal exam.
• Practical work will be worth 4 points, to be carried out during the last two weeks of classes (weeks 14 and 15), in T classes.

The student opts for continuous assessment when performing and submitting the first test for correction.

If the student does not obtain the minimum in the 2nd test, or the sum of the marks obtained in the two tests and in the practical work, is less than 9.5 values, he will immediately pass to the exam at the time of appeal. The appeal exam will have 20 values ​​and the student must obtain a grade greater than or equal to 9.5 values.

Final exam assessment:

In the evaluation by final exam, the student will have to take the exam (Normal or Resource) for 20 values ​​and will have to obtain a grade greater than or equal to 9.5 values.

NAO