Working plan

Autores: MªEugenia Rosado, Sonia Luisa Rueda

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	3	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	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

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

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

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

Week 13. Cónics III

Table of contents	Class material	Time	Methodology	Self-testing
• Determination of conics, bundles.	Week 13	3	Theory
• Determination of conics, bundles.	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	3	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.