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Working plan

Autores: MªEugenia Rosado, Sonia Luisa Rueda

 

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

 

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

 

Week 4. Diagonalization

 

Table of contents Class material   Time   Methodology Self-testing

• Eigenvalue and eigenvector.

• Eigenspaces.

• Obtainment of an eigenvector  base.

• Diagonalization.

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

2 MAPLE lab

 

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     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

 

Week 6. Affine transformations I

 

Table of contents Class material   Time   Methodology Self-testing

• Affine transformation.

• Matrix expression.

• Subspaces of fixed points. . 

  Week 6     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     Theory     
Sheet 7 2 Exercises
Mid-term exam 2 Problem solving and practice

• Exercises to determine and obtain invariant subspaces. 



2 MAPLE lab

 

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     Theory     
Sheet 8 1 Exercises

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



2 MAPLE lab

 

Week 9. Isometries

 

Table of contents Class material   Time   Methodology Self-testing

 

• Isometry.

• Matrix expression. 

• Classification

• Determination.

  Week 9     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

 

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     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     Theory     
Sheet 11 1 Exercises

• Exercises with equations of conics in different coordinate systems.



2 MAPLE lab

 

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     Theory     
Sheet 12 1 Exercises

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



2 MAPLE lab

 

Week 13. Cónics III

 

Table of contents Class material   Time   Methodology Self-testing

• Determination of conics, bundles.

  Week 13     Theory     
Sheet 13 1 Exercises

• Exercises to determine conics with bundles.



2 MAPLE lab

 

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     Theory     
Sheet 14 1 Exercises

• Exercises with quadrics in different coordinate systems.



2 MAPLE lab

 

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.

 

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