Linear Algebra

Code	14758
Year	1
Semester	S1
ECTS Credits	7,5
Workload	TP(75H)
Scientific area	Mathematics
Entry requirements	Mathematics A 12th year
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.
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.
Main Bibliography	Cabello, J. (2006). Álgebra lineal: sus aplicaciones en economía, ingenierías y otras ciencias. Madrid: Delta. Cabral, I., Perdigão, C., & Saiago, C. (2024). 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.
Language	Portuguese. Tutorial support is available in English.