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
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 |
|
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
|
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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 |
|
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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