Linear Algebra and Analytical Geometry
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2008/2009 - 1S
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
1- BACKGROUND
The promotion of logical reasoning, methods of analysis and the theoretical development of mathematical concepts is fundamental to support the study of the majority of disciplines along this course of studies.
2- SPECIFIC AIMS
This discipline aims to introduce the basic fundamental concepts of Linear Algebra, Vector Algebra and Analytic Geometry.
3- PREVIOUS KNOWLEDGE
The student must be acquainted with basic notions on trigonometry, real functions, plane analytic geometry, systems of linear equations and logic operations.
4- PERCENTUAL DISTRIBUTION
Scientific component: 100%.
5- LEARNING OUTCOMES
At the end of this, students should be capable of:
a) Knowing vector algebraic operations, their properties and how to apply them;
b) Define vector space, vector subspace and Euclidian subspace;
c) Define linear combination of vectors, linear independence and subspace spanned by a set of vectors;
d) Define a basis and dimension of vector space; obtain the coordinates of a vector with respect to a given basis;
e) Define line and plane, properties and represent lines and planes;
f) Solve problems with lines and planes, such as distances, angles and relative positions;
g) Knowing basic matrix operations, properties and operations;
h) Define and calculate the rank of a matrix;
i) Define nonsingular matrix, properties of the inverse of a matrix and calculate the inverse of a matrix;
j) Define determinant of a matrix, properties and calculate it;
k) Analyse and solve linear systems of equations;
l) Define linear transformations, define and calculate kernel and algebraic operations;
m) Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations;
n) Calculate eigenvalues and eigenvectors of linear transformations and knowing properties.
Program
Vector Algebra - The vector space of n-uples of real numbers. The dot product. Norm of a vector. Orthogonality and angle between two vectors. The linear span of a finite set of vectors. Linear independence and dependence. Bases and dimension in vector spaces. The cross product. The scalar triple product. Applications of Vector Algebra to Analytic Geometry - Lines in n-space. Properties of straight lines. Lines and vector valued functions. Linear Cartesian equations for straight lines. Planes in n-space. Properties of planes. Normal vectors to planes. Planes and vector valued functions. Linear Cartesian equations for planes. Geometric applications to three-dimensional space. Matrices - Algebraic operations. Transpose of a matrix. Square matrices: definitions and special properties. Rank of a matrix. Inverse of a square matrix. Determinants - Definition and properties. Minors and cofactors. The Laplace theorem. Computation of determinants. The determinant of the inverse of a non-singular matrix. Evaluation of the rank of a matrix with determinants. Systems of Linear Equations - Gauss and Gauss-Jordan methods. Cramer´s rule.
Linear Spaces - Definition and properties. Subspaces of a linear space. Dependent and independent sets in a linear space. Bases and dimension. Inner products. Euclidean spaces. Norms and orthogonality. Linear Transformations and Matrices - Definition. Null space and range. Nullity and rank. Algebraic operations. Inverses. One-to-one linear transformations. Matrix representation of linear transformations. Matrices representing the same linear transformation. Similar matrices. Eigenvalues and Eigenvectors - Definition and properties. Linear transformations with similar diagonal matrix representations.
Mandatory literature
Anton, Howard;
Elementary linear algebra. ISBN: 0-471-44902-4
Apostol, Tom M.;
Calculus. ISBN: 84-291-5001-3
J.A. Trigo Barbosa, J.M.A. César de Sá, A.J. Mendes Ferreira;; ALGA - Exercícios Práticos , N (Obra a adquirir na reprografia da FEUP)
Barbosa José Augusto Trigo;
Noções sobre matrizes e sistemas de equações lineares. ISBN: 972-752-069-3 972-752-065-0
J.A. Trigo Barbosa; ; ALGA - Apontamentos Teórico-Práticos , N (Obra a adquirir na reprografia da FEUP)
Complementary Bibliography
Luís, Gregório;
Álgebra linear. ISBN: 972-9241-05-8
Ribeiro, Carlos Alberto Silva;
Álgebra linear. ISBN: 972-8298-82-X
Monteiro, António;
Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
Teaching methods and learning activities
Theoretical classes: detailed exposition of the program of the discipline illustrated by application examples. Practice classes: application of the theoretical concepts in the resolution of several exercises that can be found in the proposed literature.
Evaluation Type
Distributed evaluation without final exam
Assessment Components
Description |
Type |
Time (hours) |
Weight (%) |
End date |
Subject Classes |
Participação presencial |
78,00 |
|
|
Examinations |
Exame |
3,00 |
|
2009-01-16 |
|
Total: |
- |
0,00 |
|
Amount of time allocated to each course unit
Description |
Type |
Time (hours) |
End date |
Time to study for examination |
Estudo autónomo |
24 |
2009-01-16 |
Time to study for lessons |
Estudo autónomo |
57 |
2009-01-16 |
|
Total: |
81,00 |
|