Linear Algebra and Analytic Geometry II
Objectives
At the end of this course the student will have acquired knowledge and skills that will enable him:
 To solve Linear Algebra problems using basic calculus skills and other techniques.
 To communicate, orally or by writing, knowledges, procedures, results and mathematical ideas.
 To develop the skill of identify and mathematically describe a problem, to organize the available information and to select an acquire model.
 To learn how to speak, prove and solve in Mathematics. To distinguish what are the things from how they are calculated.
 To reach a needed critical judgment to distinguish between a correct proof and other that is not.
 To start facing problems that are not exercises.
 To apply the linear álgebra techniques to other áreas, such as, to analytic geometry.
General characterization
Code
12904
Credits
6.0
Responsible teacher
António José Mesquita da Cunha Machado Malheiro
Hours
Weekly  4
Total  56
Teaching language
Português
Prerequisites
Knowledge of Linear Algebra and Analytic Geometry I.
Bibliography
 “Introdução à Álgebra Linear”, Ana Paula Santana e João Filipe Queiró 2010 Gradiva, Lisboa
 “Geometria Analítica e Álgebra Linear”, Elon Lages Lima 2015 Coleção Matemática Universitária, IMPA
 “Linear Algebra and its Applications”, Gilbert Strang 2006, 4a edição Thomson Brooks/Cole, California
Teaching method
There are classes in which theory is lectured and illustrated by examples. There are also problemsolving sessions. Some exercises are left to the students to be solved on their own as part of their learning process.
Students can ask questions during the classes, in weekly scheduled sessions or in special sessions accorded directly with the professor.
There are two midterm tests that can substitute the final exam in case of approval. Otherwise, the student must pass the final exam.
In order to be evaluated, students must obtain frequency (see details in Evaluation methods).
Evaluation method
EVALUATION RULES
LINEAR ALGEBRA AND ANALYTICAL GEOMETRY II (ALGA II)
2023/24
1. ATTENDANCE
To obtain approval for the Curricular Unit (UC) of ALGA II, the student must have attendance. In this semester, attendance is obtained through weekly forms, during the class period, and made available through the Moodle platform (called Attendance form). Each form will be available in Moodle for one week. Failure to complete more than three forms, within the established period, will result in not obtaining attendance at the UC.
2. CONTINUOUS EVALUATION
The continuous evaluation consists of carrying out, during the academic period, two inperson and/or distance tests on the Moodle platform, each one quoted from 0 to 20 values (rounded to one decimal place).
All students who, at the time of the test, are enrolled in the ALGA Curricular Unit can take any test. It is also mandatory to register for the test at CLIP.
3. APPROVAL AND FINAL CLASSIFICATION
Let T1 and T2 be the classifications obtained in the 1st and 2nd tests, respectively, rounded to decimals.
The student''''s final grade, CF, is obtained by rounding up to the units of (0.5 × T1 + 0.5 × T2), except if the CF is greater than 15, in which case the student may choose to keep the final grade of 15 or take a complementary test for grade defense. The final classification, after the defense of the note, will not exceed the classification obtained in the tests. In the defense of the grade, the student''''s performance during classes and in the activities developed in moodle will be taken into account. Approval to the UC occurs whenever CF ≥ 10. If CF is below 10 the student fails.
4. EXAM
All students enrolled in the UC who obtained attendance can take the exam  see point 1.
In this case, the exam grade (NE) replaces the test''''s grade to obtain the final grade, CF. The final grade, CF, of the student is obtained by rounding to the units of NE, except if the CF is greater than 15, in which case the student can choose between getting a final grade of 15 or taking a complementary test to defend the grade. The final classification, after the defense of the note, will not exceed the classification obtained in the exam. In the defense of the grade, the student''''s performance during classes and in the activities developed in moodle will be taken into account. Approval to the UC occurs whenever CF ≥ 10. If CF is below 10 the student fails.
5. GRADE IMPROVEMENT
All students wishing to present themselves with a grade improvement must comply with the legal registration formalities for this purpose (information at the Academic Services). The classification of the improvement exam is obtained as indicated in 4. If this result is higher than the one previously obtained in the course, it will be taken as the final grade. Otherwise, there is no improvement in the final grade.
6. TESTS AND EXAMS
In order for a student to take any of the tests (tests or exams), he/she must register with CLIP at the place and dates mentioned for this purpose.
Tests and exams will preferably be in person, and in exceptional situations, they may be carried out at a distance if this is possible. In distance tests, video surveillance means using proctoring tools will be used.
On the day of the exam, the student will have to:

Identify themselves with Identity Card or Citizen Card;

Submit a “Test Notebook” (with the header not filled in)  for facetoface tests only.
7. FINAL NOTES
It is up to the person responsible for the UC to resolve situations that are not provided for in these rules and may resort to consulting other FCT NOVA bodies for aspects where this may be justified.
The regulation was elaborated in accordance with the norms established in the FCT NOVA evaluation regulation available here.
Subject matter
1 Complex vector spaces with inner spaces. Matrix "QR" decomposition. Schur’s theorem. Normal, hermitian, real and symmetric, unitary and orthogonal matrices and its spectral theorems. Matrix similarity and Jordan normal form.
2. Quadratic forms: definition; polar form; positive, negative definite and indefinite matrices; characterization through eigenvalues; principal minors criterium. Rectangular matrix singular values decomposition. Least squares method.
3. Analytic geometry in Rn: affine space (euclidian); affine subspaces; points, line, plane, and hyperplane. Parallelism. Incidence theorems. Frames. Point coordinates with respect to a frame; Cartesian and parametric equations. Metric problems: distance and angles between affine spaces. Conics in R2 and quadrics in R3.
Programs
Programs where the course is taught: