Course: 2023/2024

Linear Algebra

(13965)

Requirements (Subjects that are assumed to be known)

Elementary notions on mathematics, including:
- Solution of linear systems (at least 3 equations)
- Evaluation of functions (in particular, of polynomial functions)
- Vectors in R^2 and R^3. Elementary euclidean geometry in R^2 and R^3 (inner product, euclidean distance, modulus of a vector)
- Basic matrix theory.

The student will acquire the basic concepts of:
1. Complex numbers.
2. Linear systems.
3. Matrix and vector algebra.
4. The determinant of a square matrix.
5. Vector subspaces in R^n and other vector spaces.
6. Eigenvalues and eigenvectors of square matrices.
7. Orthogonality and orthonormality of vectors in R^n.
The student will acquire the skills that enable them:
1. To work with complex numbers.
2. To decide about the existence and uniqueness of solutions for a system of linear equations.
3. To find, in the case when they exist, the solutions of a system of linear equations.
4. To work with vectors and matrices.
5. To compute, in the case when it exists, the inverse of a square matrix.
6. To find bases for a vector space or subspace.
7. To compute the eigenvalues and eigenvectors of a square matrix.
8. To decide whether a square matrix is diagonalizable or not.
9. To obtain an orthonormal basis from an arbitrary basis.
10. To solve least-squares problems.
11. To orthogonally diagonalize a symmetric matrix.
By the end of this content area, students will be able to have:
1. Knowledge and understanding of the mathematical principles of linear algebra underlying Electric Engineering;
2. The ability to apply their knowledge and understanding to identify, formulate and solve mathematical problems of linear algebra using established methods;
3. The ability to select and use appropriate tools and methods to solve mathematical problems using linear algebra;
4. The ability to combine theory and practice to solve mathematical problems of linear algebra.

Skills and learning outcomes

CB1. Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study.
CB2. Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study.
COCIN4. Ability to resolve problems with initiative, decision-making, creativity, and critical reasoning skills and to communicate and transmit knowledge, skills and abilities in the Industrial Engineering field.
CEB1. Ability to solve the mathematic problems arising in engineering. Aptitude for applying knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives: numerical methods; numerical algorithms, statistics and optimization.
By the end of this content area, students will be able to have:
RA1.1. Knowledge and understanding of the mathematical principles underlying their branch of engineering.
RA2.1. The ability to apply their knowledge and understanding to identify, formulate and solve mathematical problems using established methods.
RA5.1. The ability to select and use appropriate tools and methods to solve mathematical problems.
RA5.2. The ability to combine theory and practice to solve mathematical problems.

Description of contents: programme

Lecture 0. Introduction to Complex Numbers.
0.1. Definition. Sum and Product.
0.2. Conjugate, Modulus and Argument.
0.3. Complex Exponential.
0.4. Powers and Roots of Complex Numbers.
Lecture 1. Systems of Linear Equations.
1.1. Introduction to Systems of Linear Equations.
1.2. Row Reduction and Echelon Forms.
1.3. Vector Equations.
1.4. The Matrix Equation Ax=b.
1.5. Solution Sets of Linear Systems.
1.6. Linear Independence.
1.7. Introduction to Linear Transformations.
1.8. The Matrix of a Linear Transformation.
Lecture 2. Matrix Algebra.
2.1. Matrix Operations.
2.2. The Inverse of a Matrix.
2.3. Block-Partitioned Matrices.
Lecture 3. Determinants.
3.1. Introduction to Determinants.
3.2. Properties of Determinants.
Lecture 4. Vector Spaces.
4.1. Vector Spaces and Subspaces.
4.2. Null Space and Column Space of a Matrix.
4.3. Linearly Independent Sets and Bases.
4.4. Coordinate Systems.
4.5. The Dimension of a Vector Space.
4.6. Rank.
4.7. Change of Basis.
Lecture 5. Eigenvalues and Eigenvectors.
5.1. Introduction to Eigenvalues and Eigenvectors.
5.2. The Characteristic Equation.
5.3. Diagonalization of Square Matrices.
Lecture 6. Orthogonality and Least Squares.
6.1. Inner Product, Norm, and Orthogonality.
6.2. Orthogonal Sets.
6.3. Orthogonal Projections.
6.4. The Gram-Schmidt Method and the QR Factorization.
6.5. Least-Squares Problems.
Lecture 7. Symmetric Matrices.
7.1. Diagonalization of Symmetric Matrices.

Learning activities and methodology

The teaching methodology will include:
- Theory classes in large groups, where basic theoretical knowledge and skills will be presented. To facilitate their development, a textbook (''Linear Algebra and its Applications'', by David C. Lay) will be followed closely. The weekly planning of the course will be available to the students, allowing them to prepare the classes in advance.
- Solving exercises by the student, which will serve as self-assessment and to acquire the necessary skills.
- Problem solving classes in small groups, where exercises proposed to students will be explained and discussed.
- Using the electronic resources that the teacher will make available to students in the ''Aula Global'' platform.
- Tutorial sessions, individual and voluntary, in which students will have the possibility to consult the teacher their doubts and questions on the subject. The time and place of these sessions will be set by the teacher at the beginning of the course.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- David C. Lay. Linear Algebra and its Applications, 4th ed. Pearson Education, 2012..

Additional Bibliography

- B. Noble y J.W. Daniel. Álgebra lineal aplicada, 3ª Ed. Prentice Hall Hispanoamericana, 1989.
- K. Nicholson. Elementary Linear Algebra. Mc Graw Hill. 2003
- L. Spence, A. Insel y S. Friedberg. Elementary Linear Algebra. A Matrix Approach. Prentice Hall 2000.

Detailed subject contents or complementary information about assessment system of B.T.

The course syllabus may change due academic events or other reasons.