OCW - UC3M http://ocw.uc3m.es These are the search results for the query, showing results 1 to 9. Syllabus http://ocw.uc3m.es/matematicas/mathematical-methods/syllabus Programa de la asignatura: Temas que forman parte de la asignatura.
1. Systems of linear equations.

1.1. Notion of systems of linear equations.
1.2. Gaussian elimination.
1.2.1. Matrix notation.
1.2.2. Row reduction and echelon form.
1.3. Homogenous linear systems.
1.4. Applications.

2. Matrices and determinants.

2.1. Matrices.
2.1.1. Operation with matrices.
2.1.2. The inverse of a matrix.
2.1.3. Partitioned matrices.
2.1.4. LU factorization.
2.2. Determinants.
2.2.1. Properties.
2.2.2. Cramer's rule.

3. Real vector spaces.

3.1. Vector spaces and subspaces.
3.2. Null spaces and column spaces.
3.2.1. Linear transformation.
3.3. Linearly independent sets. Bases.
3.4. Dimension and rank.
3.5. Change of basis.

4. Eigenvalues and eigenvectors. Diagonalization.

4.1. Eigenvalues and eigenvectors.
4.2. Diagonalization.

5. Orthogonality and least-square problems.

5.1. Inner product, length, orthogonality.
5.2. Orthogonal projections.
5.3. The Gram-Schmidt process.
5.4. The least-squares problem.

6. The singular value decomposition.

6.1. Symmetric matrices.
6.2. Singular value decomposition.
6.3. Moore-Penrose pseudoinverse matrix.
6.4. Applications to least-square problem.

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No publisher María Barbero Liñán Full syllabus Algebra Real vector spaces Systems of linear equations Eigenvalues and eigenvectors Matrices and determinants Orthogonality and least-square problems Singular value decomposition Diagonalization Contents 2012-05-15T08:20:00Z Página
Mandatory readings http://ocw.uc3m.es/matematicas/mathematical-methods/mandatory-readings En este apartado encontraremos enlaces a ficheros (HTML, PDF, Word...) de materiales de lectura de la asignatura, listado bibliográfico o enlaces web a lecturas online.

• MR-B-001. LAY, D.C. "Linear algebra and its applications", Addison-Wesley, 4th edition, 2011.
• MR-B-002. TREFETHEN, L. N., BAU, D. "Numerical linear algebra", 1997.

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No publisher María Barbero Liñán Singular value decomposition Diagonalization Algebra Real vector spaces Systems of linear equations Eigenvalues and eigenvectors Orthogonality and least-square problems Main references Matrices and determinants 2012-05-15T08:20:00Z Página
Exercises and projects http://ocw.uc3m.es/matematicas/mathematical-methods/exercises-projects En esta sección encontramos un conjunto de ejercicios o problemas de la asignatura, trabajos o proyectos a realizar en la asignatura, estudios de casos, estudios dirigidos, ...

• EP-F-001. Homework sheet 1: Systems of linear equations. (PDFEP-F-002. Solutions. (PDF)
• EP-F-003. Homework sheet 2: Matrices and determinants. (PDF)    EP-F-004. Solutions. (PDF)   EP-F-005. Full solution to LU factorization exercises. ( PDF)
• EP-F-006. Homework sheet 3: Real vector spaces. (PDF)   EP-F-007. Solutions. (PDF)
• EP-F-008. Homework sheet 4: Eigenvalues and eigenvectors. Diagonalization. (PDFEP-F-009. Solutions. (PDF)
• EP-F-010. Homework sheet 5: Orthogonality and least-squares problems. (PDFEP-F-011. Solutions. (PDF)
• EP-F-012. Homework sheet 6: The singular value decomposition. (PDF)   EP-F-013. Solutions. (PDF)   EP-F-014.  Full solution to SVD exercises. ( PDF)

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No publisher María Barbero Liñán Singular value decomposition Diagonalization Algebra Real vector spaces Systems of linear equations Homework sheets Full solution to some exercises Short solution Matrices and determinants Orthogonality and least-square problems Eigenvalues and eigenvectors 2012-05-15T08:20:00Z Página
Evaluation tests http://ocw.uc3m.es/matematicas/mathematical-methods/evaluation-tests En esta sección encontraremos pruebas de conocimientos previos y de autoevaluación. Exámenes y sus soluciones.

• ET-E-001. Quiz 1. Units 1 and 2. (PDF). ET-E-002. Quiz 1. Solution (PDF).
• ET-E-003. Quiz 2. Units 3 and 4.(PDF).  ET-E-004. Quiz 2. Solution (PDF).
• ET-E-005. Quiz 3. Units 5 and 6.(PDF).  ET-E-006. Quiz 3. Solution (PDF).
• ET-E-007. Final exam. (PDF). ET-E-008 . Final exam. Solution (PDF).
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No publisher María Barbero Liñán Singular value decomposition Final exams Diagonalization Algebra Quizzes Unit 1-6 Eigenvalues and eigenvectors Real vector spaces Solutions Matrices and determinants Orthogonality and least-square problems Systems of linear equations 2012-05-15T08:20:00Z Página
Course Guide http://ocw.uc3m.es/matematicas/mathematical-methods/course-guide

 Unit Suggested learning time Basic lecture notes and readings Additional lectures notes and readings Homework sheets Quizzes 1 14 hours LN-F-001  (PDF). Sections 1.1, 1.2, 1.5 and 1.6 in MR-B-001. Chapters 3 and 4 in RR-B-001. Chapters 1 and 2 in RR-B-002. Section 1, 7 and 12 in RR-E-001. EP-F-001  (PDF). EP-F-002 (PDF). Exercices of sections 1.1, 1.2, 1.5 and 1.6 in MR-B-001. ET-E-001  (PDF). ET-E-002  (PDF). ET-E-007 (PDF). ET-E-008 (PDF). 2 14 hours LN-F-002 (PDF).  Sections 2.1-2.5 and chapter 3 in MR-B-  001. Chapter 1 in RR-B-001.  Chapters 3 and 6 in RR-B-  002. EP-F-003 (PDF).  EP-F-004 (PDF).  EP-F-005  ( PDF).  Exercises of sections 2.1-2.5  and chapter 3 in MR-B-001. ET-E-001 (PDF). ET-E-002 (PDF). ET-E-007 (PDF). ET-E-008 (PDF). 3 14 hours LN-F-003 (PDF).  Sections 4.1, 2.8, 4.2, 1.8, 1.9, 4.3, 4.5-  4.7, 1.7, 2.9, 5.1, 5.2 in MR-B-001. Chapter 5 in RR-B-001.  Chapter 4 in RR-B-002. EP-F-006 (PDF). EP-F-007 (PDF). Exercises of sections 4.1-4.7, 1.8, 1.9 in MR-B-001. ET-E-003 (PDF).  ET-E-004 (PDF).  ET-E-007 (PDF).  ET-E-008 (PDF). 4 14 hours LN-F-004 (PDF).  Sections 4.7, 5.1-5.4 in MR-B-001. Chapter 8 in RR-B-001.  Chapters 7.1-7.2 in RR-B-  002. EP-F-008 (PDF).  EP-F-009 (PDF).   Exercises of sections 5.1-5.4  in MR-B-001. ET-E-003 (PDF).  ET-E-004 (PDF).  ET-E-007 (PDF).  ET-E-008 (PDF). 5 14 hours LN-F-005 (PDF).  Sections 6.1-6.6 in MR-B-001. Chapters 5.1-5.6 in RR-B-  002. EP-F-010  (PDF).  EP-F-011 (PDF).  Exercises of sections 6.1-6.6  in MR-B-001. ET-E-005 (PDF).  ET-E-006 (PDF).  ET-E-007 (PDF).  ET-E-008 (PDF). 6 14 hours LN-F-006 (PDF). Sections 7.1 and 7.4 in MR-B-001. Chapters 3 and 4 in MR-B-002. Lecture notes in RR-E-001. EP-F-012 (PDF).  EP-F-013 (PDF).  EP-F-014 ( PDF).  Exercises of sections 7.1 and  7.4 in MR-B-001.  Exercises of chapters 3 and  4  in MR-B-002. ET-E-005 (PDF).  ET-E-006 (PDF).  ET-E-007 (PDF).  ET-E-008 (PDF).
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No publisher María Barbero Liñán Singular value decomposition Lecture planning Diagonalization Algebra Real vector spaces Systems of linear equations Use of materials Eigenvalues and eigenvectors Orthogonality and least-square problems Matrices and determinants 2012-05-15T08:20:00Z Página
Lecture notes http://ocw.uc3m.es/matematicas/mathematical-methods/lecture-notes En esta sección encontraremos ficheros de transparencias, de audio, de video y con orientaciones que el profesor da en clase.

• LN-F-001. Unit 1: Systems of linear equations. (PDF)
• LN-F-002. Unit 2: Matrices and determinants. (PDF)
• LN-F-003. Unit 3: Real vector spaces. (PDF)
• LN-F-004. Unit 4: Eigenvalues and eigenvectors. Diagonalization. (PDF)
• LN-F-005. Unit 5: Orthogonality and least-squares problem. (PDF)
• LN-F-006. Unit 6: The singular value decomposition. (PDF)
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No publisher María Barbero Liñán Singular value decomposition Theory Algebra Real vector spaces Systems of linear equations Eigenvalues and eigenvectors Slides Matrices and determinants Orthogonality and least-square problems Diagonalization 2012-05-15T08:20:00Z Página
Mathematical Methods II http://ocw.uc3m.es/matematicas/mathematical-methods This course consists of a introduction to linear algebra. MARÍA BARBERO LIÑÁN

Department of Mathematics

Area:
Linear algebra

Degree:
Bachelor's Degree in Statistics and Business

May, 2012

Image courtesy www.laylinalgebra.com

Lectures: 21 hours + Tutorials: 21 hours.
Total learning time (including the actual lectures): 140 hours.

#### PRERREQUISITES AND RECOMMENDED PREVIOUS KNOWLEDGE

1. Manipulate and simplify algebraic expressions.

2. Solve a system of two linear equations and two unknowns.

3. Basic knowledge of geometry and functions.

In general, the mathematics level achieved in high school and in professional colleges is enough.

#### GENERAL COURSE DESCRIPTION

Introduction to linear algebra.

#### OBJETIVES: KNOWLEDGE AND SKILLS

The student must be able to:

1. Solve and discuss systems of linear equations by Gaussian elimination.

2. Manipulate matrices and vectors (addition; multiplication, computation of inverse matrices and determinants whenever is possible).

3. Decide if a family of vectors are linearly dependent or independent.

4. Determine if a family of vectors spans a vector subspace. If so, find one of its basis.

5. Determine if a transformation is linear or not. If so, rewrite such a transformation in terms of matrices in different basis.

6. Determine if an endomorphism can be diagoanlized or not. Diagonalize endomorphisms whenever is possible.

7. Manipulate the abstract notions of scalar product and norm.

8. Obtain an orthonormal basis from a non-orthogonal basis by means of Gram-Schmidt method.

9. State and solve linear model problems by means of least square problems. Solve such problems with orthogonal projections.

10. Obtain singular value decomposition.

11. State and solve linear model problems by means of least square problems.

Moreover, the students must learn to:

1. Develop the ability to analize and summarize.

2. Model and solve problems.

3. Express and communicate mathematical reasoning in oral and written form.

#### TEACHING MATERIALS

Summary of lectures notes in slides.

Homework sheets with solutions.

Annotated bibliography based on D. Lay "Linear Algebra and its Applications", Addison Wesley, 2012.

#### ASSESSMENT ACTIVITIES OR PRACTICAL ASSIGNMENTS

In-class activities, online activities, in-class quizzes and final exam.

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No publisher María Barbero Liñán Bachelor in Statistics and Business Algebra Prerequisites Systems of linear equations Eigenvalues and eigenvectors General information Orthogonality and least-square problems Singular value decomposition Grado en Estadística y Empresa Real vector spaces Matrices and determinants Diagonalization 2012 2012-05-15T08:20:00Z Curso
Instructor http://ocw.uc3m.es/matematicas/mathematical-methods/instructor Información Profesorado ## María Barbero Liñán

Assistant professor (Ayudante doctor)

Personal web page.

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No publisher María Barbero Liñán Singular value decomposition Instructor information Algebra Real vector spaces Systems of linear equations Eigenvalues and eigenvectors Orthogonality and least-square problems Matrices and determinants Diagonalization 2012-05-15T08:16:24Z Página
Download This Course http://ocw.uc3m.es/matematicas/mathematical-methods/mathematical-methods.zip No publisher María Barbero Liñán Singular value decomposition Diagonalization Prerequisites Systems of linear equations Eigenvalues and eigenvectors General information Real vector spaces Matrices and determinants Orthogonality and least-square problems 2019-03-06T10:32:39Z Archivo