Linear Algebra and Analytic Geometry

Code:

M1004

Acronym:

M1004

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:F	65	Official Study Plan	1	-	6	56	162
			3
L:G	0	study plan from 2017/18	3	-	6	56	162
MI:EF	76	study plan from 2017/18	1	-	6	56	162

Teaching language

Portuguese

Objectives

The student should know and understand: how to solve and discuss linear systems of equations using the Gauss method through the matrix notation; some of the more important determinant properties for the calculation of the determinant of a square matrix, using them according the particularities of the matrix, and knowing the cases where area and volume interpretations are given; the basic concepts and main results on vector spaces and on linear maps between finite-dimensional linear vector spaces.

Learning outcomes and competences

The student should know: how to solve and discuss linear systems of equations;how to calculate the determinant of a square matrix,using the properties of the determinant function and its area and volume interpretations; the basic concepts and main results on finite-dimensional linear vector spaces and linear maps between finite-dimensional linear vector spaces.

Working method

Presencial

Program

• Systems of linear equations:                                                                                                                                                                  
  Equivalent systems of linear equations. Gauss method and classification of a system with respect to the number of solutions.                                                                                                                        


• 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.           

  
  


• Determinants:  
  

  Definition using the properties of a determinant. 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.                                      


• 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.

  
  

  Linear subspaces as solutions of a system of linear equations. Revisions about lines and planes through the origin in R2 and in R3.

  
  

   

  
  


• 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.

  
  

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

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Eligibility for exams

No requirements.

Calculation formula of final grade

Final exam (for the first and second exam and for students that are not trying to improve their final grade), where one group of questions (out of 2) can be replaced by the score obtained in one test, worth 10/20, about the first half of the syllabus.

In no other case will the student be allowed to replace part of the exam by a test.

Any special exam can be either an oral or a written exam. No part of these exams can be replaced by the score obtained in a test.

Examinations or Special Assignments

Any special exam can be either an oral or a written exam. No part of these exams can be replaced by the score obtained in a test.

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.
No part of these exams can be replaced by the score obtained in a test.

 

Classification improvement

Exam. For these students it will not be allowed to replace any part of the exam by any test.

Observations

Any student can be asked to do an oral examination in case there are some dougts about the written examination.