Linear Algebra and Analytical Geometry

Code:

EM0005

Acronym:

ALGA

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2010/2011 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEIG	90	Syllabus since 2006/2007	1	-	6	84	160
MIEM	221	Syllabus since 2006/2007	1	-	6	84	160

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 course units along this programme of studies.
2- SPECIFIC AIMS
This course unit 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

J.A. Trigo Barbosa; ; ALGA - Apontamentos Teórico-Práticos , N (Obra a adquirir na reprografia da FEUP)
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

Complementary Bibliography

Ribeiro, Carlos Alberto Silva; Álgebra linear. ISBN: 972-8298-82-X
Monteiro, António; Álgebra linear e geometria analítica. ISBN: 972-8298-66-8
Luís, Gregório; Álgebra linear. ISBN: 972-9241-05-8

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
Attendance (estimated)	Participação presencial	78,00
Examinations	Exame	3,00		2011-02-07
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Time to study for examinations	Estudo autónomo	24	2011-02-07
Time to study for lessons	Estudo autónomo	57	2011-02-07
	Total:	81,00

Eligibility for exams

Admission to exams
1) Not exceed the absence limit allowed in Article 4 of the General Evaluation Rules of FEUP
2) attend to at least one of the two tests with minimal classification of 4 out of 20.

Calculation formula of final grade

The student must attend to two written exams, with the duration of 1 hour and 30 minutes each. Each exam comprises two different parts: a theoretical part, which worth 20% of the final mark and a theoretical-practical part which worth 80% of the final mark.

Exams are scheduled for these dates:
1st written exam: 23 November 2009
2nd written exam: 18 January 2009

Final mark will be based on the average grade of the two exams.
According to Article 8 of General Evaluation Rules of FEUP, a student in order to pass the course must earn a grade of six out of twenty or better in each of the exams.
At the end of the Semester students will be able to attend a new exam in order to improve their final grade. This exam may either only test a part of the program, or the whole program.
Only students who did not exceed the absence limit allowed can attend this exam.
Date of the exam: 8 February 2010.

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to items 6 and 7 in Article 6 of General Evaluation Rules of FEUP.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to General Evaluation Rules of FEUP.