Course: 2019/2020

Linear Algebra

(13402)

Competences and skills that will be acquired and learning results. Further information on this link

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 Rn and other vector spaces.
6. Eigenvalues and eigenvectors of square matrices.
7. Orthogonality and orthonormality of vectors in Rn.
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.

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, 4th edition) will be followed closely. The chronogram 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 Edition. Prentice-Hall. 2012

Additional Bibliography

- B. Noble, J.W. Daniel,. Applied Lineal Algebra. Prentice. 1977
- 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