Perfilado de sección

    • Week 1. Vector spaces I

      Week 2. Vector spaces II

      Week 3. Linear transformations

      Week 4. Diagonalization

      Week 5. Affine space

      Week 6. Affine transformations I

      Week 7. Affine transformations II

      Week 8. Euclidean affine space

      Week 9. Isometries

      Week 10. Introduction to projective space

      Week 11. Cónics I

      Week 12. Cónics II

      Week 13. Cónics III

      Week 14. Quadrics I

      Week 15. Quadrics II

       

       

      Week 1. Vector Spaces I
      Table of contents Class material   Time   Methodology Self-testing

       • Definition.

      • Subspaces.

      • Linear combination of vectors.

      • Generated subspaces.

      • Linear dependency and independency.

      • Base and dimension.

      		  Week 1       3   Theory       
      Sheet 1 1 Exercises
      • Overview of rank, determinant, and their application to the study of linear dependency of vectors.

      		2 MAPLE lab

       

      Week 2. Vector Spaces II
      Table of contents Class material   Time   Methodology Self-testing

       • Vector coordinates.

      • Intersection of subspaces.

      • Sum of subspaces.

      • Equations of subspaces.

      		   Week 2       3   Theory        
      Sheet 2 1 Exercises
      • Overview of methods to resolve systems of equations.

      		2 MAPLE lab
      Operation subspaces (PDF)

       

      Week 3. Linear transformations
      Table of contents Class material   Time   Methodology Self-testing

      • Linear transformation.

      • Matrix expression.

      • Kernel and image.

      • Operations with linear transformations.

      • Change of base.

      		    Week 3      3  Theory         
      Sheet 3 1 Exercises

      • Exercises to determinate linear transformations, images.

      • Discussion and obtainment of the origin of a vector.

      		 2 MAPLE lab
      Linear maps (PDF)

       

      Week 4. Diagonalization
      Table of contents Class material   Time   Methodology Self-testing

      • Eigenvalue and eigenvector.

      • Eigenspaces.

      • Obtainment of an eigenvector  base.

      • Diagonalization.

      		   Week 4     3  Theory        
      Sheet 4 1 Exercises
      • Exercises about diagonalization.

      		2 MAPLE lab
      Diagonalization (PDF)

       

      Week 5. Affine space
      Table of contents Class material   Time   Methodology Self-testing

      • Affine space.

      • Affine subspace. 

      • Dimension. 

      • Coordinate systems, change of coordinate systems. 

      		   Week 5     3  Theory     
      Sheet 5 1 Exercises

      • Exercises about equations of subspaces.

      • Equations of subspaces in different coordinate systems, in dimensions 2 and 3

      • Examples of sum and intersection of subspaces.

      		2 MAPLE lab
      Affine space (PDF)

       

      Week 6. Affine transformations I
      Table of contents Class material   Time   Methodology Self-testing

      • Affine transformation.

      • Matrix expression.

      • Subspaces of fixed points. . 

      		   Week 6     3  Theory     
      Sheet 6 1 Exercises

      • Exercises to determine affine transformations.

      • Exercises about homotheties, oblique symmetries and projections. 

      		2 MAPLE lab

       

      Week 7. Affine transformations II
      Table of contents Class material   Time   Methodology Self-testing

      • Invariant subspaces in affine transformations.

      		    Week 7     3  Theory     
      Sheet 7 2 Exercises
      Mid-term exam 2 Problem solving and practice

      • Exercises to determine and obtain invariant subspaces. 

      		2 MAPLE lab
      Isometries E2 (PDF)

       

      Week 8. Euclidean affine space
      Table of contents Class material   Time   Methodology Self-testing

      • Affine euclidean space.

      • Orthogonal and orthonormal coordinate systems. 

      • Orthogonal matrices.

      		   Week 8     3  Theory     
      Sheet 8 1 Exercises

      • Exercises about changing from an orthonormal base to an orthogonal one.

      		2 MAPLE lab
      Rotation E3 (PDF)

       

      Week 9. Isometries
      Table of contents Class material   Time   Methodology Self-testing

       

      • Isometry.

      • Matrix expression. 

      • Classification

      • Determination.

      		   Week 9     3  Theory     
      Sheet 9 1 Exercises

      • Exercises to determinate and obtain invariant subspaces.

      • To sum up affine transformations and isometries, examples of similarities.

      		2 MAPLE lab
      Symmetries E3 (PDF)

       

      Week 10. Introduction to projective space
      Table of contents Class material   Time   Methodology Self-testing

      • Projective plane and space.

      • Projectivized affine space.

      • Homogeneous coordinates.

      • Conics will be introduced as plane sections of a cone.

       
      		   Week 10     3  Theory     
      Sheet 10 1 Exercises

      • Exercises about equations of lines and planes in the projective space.

      		2 MAPLE lab

       

      Week 11. Cónics I
      Table of contents Class material   Time   Methodology Self-testing

      • Definition as locus.

      • Optic propieties.

      • Reduced equations.

      • Equation in the projective space.

      • Matrix expression.

      • Intersection with a line, tangents.

      • Intersection with the line at infinity. Affine classification.

      		   Week 11     3  Theory     
      Sheet 11 1 Exercises

      • Exercises with equations of conics in different coordinate systems.

      		2 MAPLE lab
      Hyperbola (PDF)

       

      Week 12. Cónics II
      Table of contents Class material   Time   Methodology Self-testing

       

      • Harmonic quatern.

      • Pairs of conjugated points.

      • Polarity.

      • Singular points, degenerated conics.

      • Notable elements.

      		   Week 12     3  Theory     
      Sheet 12 1 Exercises

      • Exercises about polarity, obtainment of notable elements, center, diameters, aymptotes, axes.

      		2 MAPLE lab
      Parabola (PDF)

       

      Week 13. Cónics III
      Table of contents Class material   Time   Methodology Self-testing

      • Determination of conics, bundles.

      		   Week 13     3  Theory     
      Sheet 13 1 Exercises

      • Exercises to determine conics with bundles.

      		2 MAPLE lab
      Determining conics (PDF)

       

      Week 14. Quadrics I
      Table of contents Class material   Time   Methodology Self-testing

      • Geometric introduction.

      • Reduced equations.

      • Equation in the projective space.

      • Matrix expression.

      • Intersection with a line, tangent lines and planes.

      • Intersection with the point at infinity, affine classification.

      • Singular points, degenerate conics.

      • Notable elements

      		   Week 14     3  Theory     
      Sheet 14 1 Exercises

      • Exercises with quadrics in different coordinate systems.

      		2 MAPLE lab
      Quadrics classification (PDF)

       

      Week 15. Quadrics II
      Table of contents Class material   Time   Methodology Self-testing

      • Polarity.

      • Singular points, degenerate quadrics.

      • Notable elements.

       
      		 Week 15       1.5   Clase teoríca      Método Expositivo    
      Sheet 15 5
      Explicación de contenidos



      • Exercises about polarity, obtainment of notable elements, center, diameters, diametral planes, axis.