Linear Algebra II
Code
10973
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
9.0
Teacher in charge
Júlia Maria Nunes Loureiro Vaz de Carvalho
Weekly hours
6
Teaching language
Português
Objectives
The student is supposed to consolidate and complement knowledge acquired in Linear Algebra I (vide syllabus). In learning process, logical reasoning and critical mind should continue being developed.
Prerequisites
Knowledge corresponding to the contents of Linear Algebra I (1st semester-1st year)
Knowledge about inner product spaces. The necessary notions and results constitute the first part of the syllabus of the unit Geometry.
Subject matter
1. Eigenvalues and eigenvectors of endomorphisms and matrices – Definitions and properties. Eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicities. Diagonalization. Cayley-Hamilton theorem. Minimum polynomial.
2. Endomorphisms of inner product spaces (finite dimension) – Adjoint of an endomorphism; normal endomorphism, hermitian (symmetric), skew-hermitian (skew-symmetric), unitary (orthogonal) and respective definitions for square matrices. Positive definite endomorphism, positive semidefinite, negative definite, negative semidefinite, indefinite and respective definitions for square matrices. Relationship between the different types of endomorphisms and the respective matrices determined by an orthonormal basis. Fundamental results involving these notions, in particular, Schur theorem and spectral theorem.
3. Jordan canonical form and some of its fundamental consequences.
Bibliography
1. Apostol, T. M., Linear Algebra – a first course with applications to differential equations, John Wiley & Sons, 1997.
2. Anton, H., e Rorres, C., Elementary Linear Algebra - Applications Version, 9th Edition, John Wiley & Sons, 2005.
3. Friedberg, S.H., Insel, A. J., e Spence, L. E., Linear Algebra, 3rd Ed., Prentice Hall, 1997.
4. Horn, R. A., e Johnson, C. R., Matrix Analysis, Cambridge University Press, 1985.
5. Leon, S. J., Linear Algebra with Applications, 7th Ed., Prentice Hall, 2006.
6. A. P. Santana, J. F. Queiró, Introdução à Álgebra Linear, Gradiva, 2010.
Teaching method
There are classes in which theory is lectured and illustrated by examples. There are also problem-solving sessions. For each chapter there is a list of proposed exercises that the students should solve. Most of the exercises is corrected in the problem-solving sessions.
Evaluation method
Students enrolled for the first time in the unit must attend all classes, except up to 3 lectures and up to 3 problem-solving classes.
Students that have already been enrolled in the unit must attend, at least, 2/3 of the lectures and 2/3 of the problem-solving classes.
The students that do not fulfill the above requirements automatically fail "Álgebra Linear II".
There are three mid-term tests. These tests can substitute the final exam if the student has grade, at least, 7.5 in the third one and CT is, at least, 9.5. CT is calculated as follows:
CT = 0,30*T1 + 0,35*T2 + 0,35*T3
where Ti, 1 ≤ i ≤ 3, is the non-rounded grade obtained in test i.
If the student satisfies the conditions above with CT (rounded to units) greater than 16, he may choose between having 16 as final grade or undertake a complementary assessment.
To be approved in final exam, the student must have a minimum grade of 9.5 in it. Again, for grades (rounded to units) greater than 16, the student must undertake a complementary assessment, otherwise his final grade will be 16.
More detailed rules are available in the portuguese version.
The non-portuguese students are advised to address the professor for more detailed information.