Course: 2023/2024

Linear Algebra

(15488)

The student will acquire appropriate skills for:
1. Operating and solving simple equations with complex numbers
2. Computing the solution of a linear system of equations
3. Deciding the uniqueness and existence of the solution of a linear system of equations
4. Working with vectors and matrices
5. Computing the inverse of a matrix
6. Computing bases of linear subspaces
7. Computing eigenvalues and eigenvectors of a matrix
8. Computing an orthonormal basis
9. Solving least square problems
10. Diagonalizing symmetric matrices through orthogonal transformations.
Students that pass this subject successfully will have:
1. Knowledge and understanding of the mathematical principles of algebra underlying their branch of engineering;
2. The ability to apply their knowledge and understanding to identify, formulate and solve mathematical problems using established methods of algebra;
3. The ability to select and use appropriate tools and methods of algebra to solve mathematical problems;
4. The ability to combine theory and practice of algebra to solve mathematical problems

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.
CG1. Ability to solve problems with initiative, decision-making, creativity, critical reasoning and to communicate and transmit knowledge, skills and abilities in the field of Industrial Engineering.
CG11. Ability to solve mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential and partial derivative equations; numerical methods; numerical algorithms; statistics and optimisation.
RA1. Knowledge and understanding: Have basic knowledge and understanding of science, mathematics and engineering within the industrial field, as well as knowledge and understanding of Mechanics, Solid and Structural Mechanics, Thermal Engineering, Fluid Mechanics, Production Systems, Electronics and Automation, Industrial Organisation and Electrical Engineering.
RA2. Engineering Analysis: To be able to identify engineering problems within the industrial field, recognise specifications, establish different resolution methods and select the most appropriate one for their solution
RA5. Engineering Applications: To be able to apply their knowledge and understanding to solve problems and design devices or processes in the field of industrial engineering in accordance with criteria of cost, quality, safety, efficiency and respect for the environment.

Description of contents: programme

1. Complex numbers
· Numbers sets
· Necessity of complex numbers
· Binomial form of a complex number
· Graphical representation
· Operations
· Complex conjugate, modulus, argument
· Polar form of a complex number
· Roots of complex numbers
· Exponential of a complex number
· Solving equations
2. Systems of linear equations
· Introduction to Linear Equations
· Geometrical Interpretation
· Existence and Uniqueness
· Matrix Notation
· Gaussian Elimination
· Row Equivalence and Echelon Forms
· Solving Linear Systems
· Homogeneous Systems
· Simultaneous Solving
· Systems with parameters
3. The vector space Cn
· Vectors
· Linear Subspace
· Linear Combinations
· Subspace Spanned by Vectors
· Column and Row Spaces
· The Matrix Equation Ax=b
· Null Space
· Revisiting Linear Systems
· Linear Independence
· Basis for a Linear Subspace
· Dimension of a Linear Subspace
· Basis for Col A, Row A and Nul A
· Rank of a Matrix
· Coordinate Systems
· Introduction to Linear Transformations
4. Matrix algebra
· Matrix Operations
· Transpose of a Matrix
· Conjugate Transpose of a Matrix
· Inverse of a Matrix
· Partitioned Matrices
· Determinants
5. Eigenvalues and eigenvectors
· Eigenvalues & Eigenvectors
· The Characteristic Equation
· Diagonalization
· Change of Basis
· Transformations between Linear Subspaces
6. Orthogonality
· Dot Product and Modulus
· Orthogonal Sets
· Unitary Matrices
· Orthogonal Complement
· Orthogonal Projection
· The Gram-Schmidt Process
· The QR decomposition
· Least-Squares Problems
7. Normal matrices
· Schur Decomposition
· Normal Matrices & Unitary Diagonalization
· Particular Cases of Normal Matrices

Learning activities and methodology

- Theory classes in large groups, which could be given in "flipped classroom" format. In them, basic theoretical knowledge and skills will be presented. A textbook will be followed (Linear Algebra and Its Applications, by David C. Lay). The course chronogram will be available in advance to the students to allow them to prepare the classes in advance.
- Problem solving classes in small groups.
- Solving problems by the student.
- The teacher may propose additional homework to be done either individually or in group.

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. Pearson, 2012.

Additional Bibliography

- B. Noble and J. W. Daniel. Applied Linear Algebra, 3rd Edition. Prentice Hall, 1988.
- G. Strang.. Linear Algebra and its applications, 4th ed.. Wellesley, Cambridge,. 2006
- L. Spence, A. Insel y S. Friedberg. Elementary Linear Algebra. A Matrix Approach. Prentice Hall 2000.

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