| Code |
15068
|
| Year |
1
|
| Semester |
S1
|
| ECTS Credits |
6
|
| Workload |
TP(60H)
|
| Scientific area |
Mathematics
|
|
Entry requirements |
Mathematics 12(A)
|
|
Learning outcomes |
After successfully completing the course, students will have developed the following skills:
(a) perform operations with matrices and solve systems of linear equations;
(b) calculate determinants and use them to solve problems;
(c) determine vector spaces and subspaces, linear combinations and generating sets, linear dependence and linear independence, basis and dimension of a vector space;
(d) define linear transformations and determine the matrix of a linear transformation;
(e) calculate eigenvalues and eigenvectors of a given matrix and diagonalize a matrix (if possible);
(f) determine change of base matrix and solve related problems;
(g) interpret and solve problems related to the dot product, norm, vector product and mixed product of vectors,
orthogonalization.
(h) use the concepts learned in this course unit to solve problems.
|
|
Syllabus |
1. Matrices and Systems of Linear Equations
Types of matrices; matrix and vector operations; elementary operations and Gaussian elimination; Gauss method and Gauss-Jordan elimination methods for solving systems of linear equations; inverse matrices
2. Determinants
Definition and properties; adjoint matrix and inverse matrix, applications.
3. Vector Spaces
Vector space and vector subspace, linear combinations and generating sets, linear dependence and linear independence, basis and dimension of a vector space.
4. Linear Transformations
Definition and properties, matrix of a linear transformation,change of basis matrix.
5. Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of a matrix,matrix diagonalization.
6. Inner Product Spaces
Inner products, norm, projection,orthonormal bases; Gram-Schmidt orthogonalization process, orthogonal complement of subspace. Vector product,mixed product; geometric applications in R3
7. Normed spaces
Vector and matrix norms
|
|
Main Bibliography |
- Isabel Cabral, Cecília Perdigão, Carlos Saiago, Álgebra Linear, Escolar Editora, 2009
- Luís T. Magalhães, Álgebra Linear como introdução à Matemática Aplicada, Escolar Editora, 2001
- David C. Lay, Linear Algebra and its applications, 5th edition, Pearson, 2016
- Gilbert Strang, Linear Algebra and its applications, 4th edition, Brooks Cole, 2005
|
|
Teaching Methodologies and Assessment Criteria |
The classes combine theory with practice. The teacher introduces the concepts, states and proves the fundamental results, provides examples and applications. The students are encouraged to participate in the classes, to interact with the teacher and colleagues and to work independently, by solving exercises, guided reading, problem formulation and problem solving.
The evaluation carried out during the teaching-learning process consists of two written tests.
|
|
Language |
Portuguese. Tutorial support is available in English.
|