Fichas de Asignaturas Comunes

 

Course Datasheet

Course	Linear Algebra
ACADEMIC YEAR:   2012/2013	2009/20102010/20112011/20122012/2013
Credits	6
Laboratory Classes	4
Coordinator/Teachers	Ricardo Monedero Rafael José Hernández(1º Periodo) Gerardo Pérez(1º Periodo) Juana Sendra(1º Periodo)
Course Area	Mathematics
Programmes
Course Syllabus	Course - Autumm Course - Spring

1. Competencies

The objective of this subject is to allow the student to develop the following generic competencies:

• Skilled to searching and selecting information, critical reasoning and writing and defending the reasonings within the defined area.
• Skilled for public speaking and in written and communicating information throughout documents and public speeches.
• Skilled for abstration, analysis and synthesis and problem solving.
• Skills for the use of Information Technologies and Communications.

And the following specific competencies (Ministerial Order CIN/352/2009, were the necessary requirements to be qualified to practice as a Telecommunications technical engineer are established):

• Capacity of solving mathematic problems that can appear in engineering. Aptitude for applying knowledges about: linear algebra, geometry, differential geometry, differential and integral calculus, differential equations, partial-differential equations, numeric methods, numeric algorithmics, statistics and optimization.

2. Learning outcomes

To develop the skills previously listed above, students must achieve the following learning outcomes:

• Perform basic operations on matrices. Computation of the inverse matrix.
• Manage the concepts of norm, distance and angle in Euclidian spaces and the orthogonalization of systems of vectors.
• "Solve linear ODEs with constant coefficients using the related theory and methods like ""undetermined coefficients"" or ""variation of constants""."
• Use tools to make calculations and to solve problems related to the subject.
• Model and solve physical problems by differencial equations systems.
• Propose and solve problems in the field of physics and engineering using linear ODEs. In particular the second orden ODE as a RLC circuit model.
• Use the concept of linear transformation between vector spaces.
• Get the diagonal form of a diagonalizable square matrix.
• Obtain the eigenvalues and eigenvectors of a square matrix.
• Use the vector space structure and its main properties.
• Represent and manage basis changes.
• Represent in a matrix form a linear transformation respect different basis.
• Compute determinants of square matrices using different techniques.
• Solve and discuss systems of linear equations using several methods.
• Obtain the LU factorization of a matrix.

3. Contents

The training activities that will be conducted in this subject are structured in the following thematic units:

Unit1.	SYSTEMS OF LINEAR EQUATIONS.
1.1.	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
Unit2.	MATRIX ALGEBRA AND DETERMINANTS.
2.1.	Matrix operations.
2.2.	Inverse of a matrix.
2.3.	Characterization of invertible matrices.
2.4.	LU factorization.
2.5.	Determinants and properties.
2.6.	Cramer's rule.
Unit3.	VECTOR SPACE AND LINEAR APPLICATIONS.
3.1.	Real vector spaces and subspaces.
3.2.	Linear dependence, rank, dimension and base.
3.3.	Linear Transformations: kernel, image, associated matrix,
3.4.	Applications composition and reverse application.
3.5.	Coordinate systems. Change of basis.
Unit4.	DIAGONALIZATION.
4.1.	Eigenvalues and eigenvectors.
4.2.	The Characteristic Equation. Eigenspace.
4.3.	Diagonalization.Endomorphisms.
Unit5.	ORTHOGONALITY AND LEAST SQUARES.
5.1.	Inner product, length and norm.
5.2.	Orthogonal sets.
5.3.	Orthogonal projections.
5.4.	The Gram-Schmidt Process.
5.5.	Least-squares problem and applications.
Unit6.	DIAGONALIZATION OF SYMMETRIC MATRICES.
6.1.	Orthogonal diagonalization.
Unit7.	LINEAR DIFFERENTIAL EQUATIONS OF HIGHER ORDER.
7.1.	Order 2Resolution linear differential equations.
7.2.	Order n.Resolution linear differential equations.
7.3.	Systems of Linear Differential Equations.