| Code |
14758
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
7,5
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| Workload |
TP(75H)
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| Scientific area |
Mathematics
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Entry requirements |
Mathematics A 12th year
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Learning outcomes |
At the end of this course, students will be able to:
- identify square and rectangular matrices, rows, and columns of a matrix,
- identify diagonal, symmetric, skew-symmetric and hermitian matrices;
- compute the sum, the product and the transpose of a matrix;
- compute the rank of a matrix;
- identify an invertible matrix and compute its inverse;
- solve and classify linear systems of equations by Gauss elimination;
- compute the determinant of a matrix;
- identify subspaces of a vector space and determine a basis;
- compute the matrix of a linear transformation;
- solve linear systems of equations and compute the inverse of a matrix by using determinants;
- compute the eigenvalues of a matrix and identify diagonalizable matrices;
- compute the inner product, the vector cross product and the scalar triple product of vectors;
- apply the Gram-Schmidt orthogonalization process.
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Syllabus |
0. Motivation
1. Matrices and systems of linear equation
Types of matrices;
Operations with matrices;
The rank of a matrix;
The inverse of a matrix;
Systems of linear equations.
2. Determinants
Determinant of a square matrix;
Properties; Cofactors; Laplace’s Theorem;
Adjoint matrix and inverse matrix;
Application to linear systems of equations;
3. Vector Spaces
Definition of vector space;
Subspaces; Linear combinations and spanning set;
Linear dependence and independence;
Basis and dimension of a vector space;
4. Linear Transformations
Definition and examples;
Properties;
Matrix of a linear transformation;
Change of basis matrix;
5. Eigenvalues and eigenvectors of a matrix
Definition, examples, and properties;
Similar matrices;
Diagonalizable matrices;
6. Inner product spaces
Inner product,
Norm;
Cauchy–Schwarz inequality;
Orthogonality, orthonormal basis and Gram-Schmidt orthogonalization process;
Ortogonal decomposition;
Vector cross product in R3 and scalar triple product.
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Main Bibliography |
Cabello, J. (2006). Álgebra lineal: sus aplicaciones en economía, ingenierías y otras ciencias. Madrid: Delta.
Cabral, I., Perdigão, C., & Saiago, C. (2009). Álgebra linear: teoria, exercícios resolvidos e exercícios propostos com
soluções. Lisboa: Escolar.
Dias Agudo, F. R. (1996). In trodução à Álgebra Linear e Geometria Analítica. Lisboa: Escolar.
Howard, A., & Busby, R. (2006). Álgebra Linear Contemporânea. Porto Alegre: Bookman.
Lay, D. C. (2007). Álgebra Linear e as suas aplicações. Rio de Janeiro: LTC.
Lipschutz, S. (1972). Álgebra linear: resumo da teoria. São Paulo: McGraw-H ill.
Magalhães, L. T. (2001). Álgebra linear como introdução a matemática aplicada. Lisboa: Escolar.
Nering, E. D. (1970). Linear Algebra and Matrix Theory. New York: John Wiley.
Strang, G. (1976). Linear Algebra and Its Applications. New York: Academic.
Santana, A. P., & Queiró, J. F. (2010). Introdução à Álgebra Linear. Lisboa: Gradiva.
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Teaching Methodologies and Assessment Criteria |
Each class (TP), which is a combination of a lecture and a practical class, is planned not only for the presentation of theoretical concepts of the syllabus by the teacher but also to enhance task solving activities by the students under the supervision of the teacher. In this way, students are expected to autonomously acquire and apply the concepts. Thus, assessment is particularly relevant to allow the student, throughout the semester, to show the skills acquired through his work at the different stages. To do so, three written tests will take place.
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Language |
Portuguese. Tutorial support is available in English.
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