Linear Algebra and Analytic Geometry I
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2021/2022 - 1S ![Requerida a integração com o Moodle Ícone do Moodle](/fcup/pt/imagens/MoodleIcon)
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
Understanding and ability to use the basic concepts and results related to the subjects of the syllabus.Learning outcomes and competences
By completing this course, students should know, understand and be able to use the basic notions and results about vector spaces; Vector subspaces; subspace sums; direct sums of subspaces; linear independence; generating systems;finitely generated vector spaces; bases; dimension; linear maps; kernel and image of linear maps; inverse image of an element; characteristic of a linear transformation; matrices; matrix of a linear map with respect to fixed bases; change of basis; application of these concepts and results to solve systems of linear equations; Similar matrices; determinants; real Euclidean spaces; inner product, norm; angle between two vectors; vector product in R3; orthonormal bases; orthogonal complement; orthogonal projection.
Working method
Presencial
Program
1. Systems of linear equations. Equivalent systems of linear equations. Gauss method and classification of a system with respect to the number of solutions.
2. Matrices: Square matrices, diagonal, upper and lower triangular matrices. Addition of matrices and product of a matrix by a scalar. The identity matrix. The rank of a matrix. Matrix form of a linear system and classification of the system using the rank of the matrix. Invertible (square) matrices. Uniqness of the inverse. The inverse of the product of two invertible matrices. The rank of an invertible matrix. How to compute the inverse of a matrix.The transpose of a matrix and the transpose of the product of two matrices.
3. Determinants. Definition, properties and examples. Relation between the determinant and elementary operations on the rows of the matrix. Relation between the rank of a matrix being maximal and the determinant being non-zero. Laplace theorem and Sarrus Rule. The adjoint matrix, its determinant and how to compute the inverse of a matrix using the adjoint. Cramer systems and their solutions. Cross product and its geometrical interpretation. The absolute value of the determinant of a 2x2 matrix resp. a 3x3 matrix.
4. Vector spaces. Definition, properties and examples. Vector subspaces, description of the ones of R2 and of R3.
Sum and intersection of vector subespaces. The vector space of solutions of an homogeneous linear system. Construction of solutions of a linear system using the solutions of the associated homogeneous linear system. Linear combination. Subspace generated by a set of vectors. Linear and independence linear. Steinitz exchange lemma. The concept of basis and dimension of a vector space. Relation between the dimension of the subspace generated by vectors in Rn and the rank of the matrix formed by these vectors. Line and column rank. Ordered basis and coordinates with respect to a basis. Invertibility of a change of basis matrix. Direct sums of subspaces and its dimension. Complementary subspaces.
Subspaces as solutions of a system of linear equations. Revisions about lines and planes through the origin in R2 and in R3.
5. Linear maps. Definition and examples. The image and the inverse image of a subspace by a linear map. Defining a linear map using only the images of the elements of o basis. Matrix of a linear map. Relation between two matrices of the same linear map with respect to different basis. The composition of linear matrices and its matrix. Isomorphisms. Description of the matrix of the inverse of an isomorphism. The kernel of a linear map. The Kernel of a monomorphism. The theorem of dimensions. Indentification, up to isomorphim, of a finite dimensional vector space over K with Kn. Dimension of the vector space of all linear maps between finite dimensional spaces. Identificantion between matrices and linear maps between finite dimensional vector spaces. Determinant of an endomorphism of a finite dimensional vector space.
6. Real Euclidean spaces; inner product, norm; angle between two vectors; vector product in R3; orthonormal bases; orthogonal complement; orthogonal projection.Mandatory literature
Anton Howard;
Elementary linear algebra. ISBN: 0-471-66959-8
Edwards jr. C. H.;
Elementary linear algebra. ISBN: 0-13-258245-7
Monteiro António;
Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
Mansfield Larry E.;
Linear algebra with geometric applications. ISBN: 0-8247-6321-1
Ana Paula Santana;
Introdução à álgebra linear. ISBN: 978-989-616-372-3
Teaching methods and learning activities
Contact hours are divided into theoretical and theoretical-practical. The former consist of lectures on the contents of the syllabus, making use of examples to illustrate the concepts treated and to guide students. In the latter, theoretical and practical exercises and problems are solved. Support materials are available on the course page. In addition to the classes, there are weekly periods where students have the opportunity to ask for help on their difficulties.keywords
Physical sciences > Mathematics > Geometry
Physical sciences > Mathematics > Algebra
Evaluation Type
Distributed evaluation without final exam
Assessment Components
designation |
Weight (%) |
Teste |
100,00 |
Total: |
100,00 |
Amount of time allocated to each course unit
designation |
Time (hours) |
Estudo autónomo |
159,00 |
Frequência das aulas |
84,00 |
Total: |
243,00 |
Eligibility for exams
No requirements.
Calculation formula of final grade
">Assessment will be based on two tests (the last one will take place on the date of the "exame da época normal"), worth 8 and 12 points and on 5 quizzes.
The first test will take 75 minuts and the second no more than 110 minuts (to be confirmed). The first test will be on day indicated at the begining of classes and the second on the day of "exame da época normal".
The 5 quizzes are together worth 1.5 points (so 0.3 each). The dates for the quizzes will be combined at the begining of classes, the quizzes will done using multiple choice quizzes in Moodle and/or short answers questions.
If T1 is the score of the first test, T2 the score of the second and Q the sum of the scores obtained in the 5 quizzes, a student will pass the course if
T1+T2 ≥ 8.5 e T1+T2+Q ≥ 9.5
and the final mark is
min{ T1+T2+Q, 20}.
In case presencial tests are not allowed in the FCUP during the first semester (for instance because of the covid-19) , the 2 tests will be done on the same day - the day schedule for the "exame da época normal". In such a case, the duration of the tests will be adjusted acording to the rules imposed by the faculty (FCUP).
">There will be an exam (época de recurso), accessible to any student who has not passed in the regular season or wants to improve the score.
Special assessment (TE, DA, ...)
Any examination required under special statutes will consist of a written exam which may be preceded by a previous oral test. If the in the oral test the students shows that he doesn't know enough, the student will be failed and can not go to the written examination.
Classification improvement
Exam. These students will not the allowed to take part of the continuous assessment of the course.
Observations
Any student can be asked to do an oral examination in case there are some doubts about the written examination.