Linear Algebra and Analytical Geometry

Code:

EM0005

Acronym:

ALGA

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2007/2008 - 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
LEM	0	Plano de estudos de transição para 2006/07	1	6	6	84	160
LGEI	0	Plano de estudos de transição para 2006/07	1	6	6	84	160
MIEIG	68	Plano de estudos de transiçao para 2006/07	1	-	6	84	160
		Syllabus since 2006/2007	1	-	6	84	160
MIEM	246	Syllabus since 2006/2007	1	-	6	84	160
		Plano de estudos de transição para 2006/07	1	-	6	84	160

Teaching language

Portuguese

Objectives

SPECIFIC AIMS:
This discipline has two main objectives: the promotion of logical reasoning and methods of analysis and the introduction and theoretical development of a set of concepts that will be fundamental to support the study of other disciplines along this course of studies.

LEARNING OUTCOMES:
At the end of this, students should be capable of:
1) Knowing vector algebraic operations, their properties and how to apply them;
2) Define vector space, vector subspace and Euclidian subspace;
3) Define linear combination of vectors, linear independence and subspace spanned by a set of vectors;
4) Define a basis and dimension of vector space; obtain the coordinates of a vector with respect to a given basis;
5) Define line and plane, properties and represent lines and planes;
6) Solve problems with lines and planes, such as distances, angles and relative positions;
7) Knowing basic matrix operations, properties and operations;
8) Define and calculate the rank of a matrix;
9) Define nonsingular matrix, properties of the inverse of a matrix and calculate the inverse of a matrix;
10) Define determinant of a matrix, properties and calculate it;
11) Analyse and solve linear systems of equations;
12) Define linear transformations, define and calculate kernel and algebraic operations;
13) Define change-of-basis matrix and apply it to problems with vector spaces and linear transformations;
14) 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

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)
Anton, Howard; Elementary linear algebra. ISBN: 0-471-44902-4

Teaching methods and learning activities

Theoretical classes: detailed exposition of the program of the discipline illustrated by application examples. Theoretical-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		2008-01-18
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Study and problems solving	Estudo autónomo	81	2008-01-18
	Total:	81,00