Linear Algebra (2500001) – Course 2025/26 PDF
Contents
Definition and examples. Vector subspaces. Linear dependence and independence. Generator system. Spaces of finite dimension. Bases. Range of a vector system. Intersection and sum of subspaces. Direct sum. Grassmann Basic problems Solving problems of dependence and linear independence. Calculation of bases. Component calculation. Problems of intersection and sum of subspaces.
Specific Objectives
Vector spaces constitute the general framework of applications in engineering and not just three-dimensional space.
Dedication
6h Large group + 5h Medium group + 15h 24m Self StudyDefinitions and types. Sum of matrices.Scalar product- Matrix product. Elementary row and column operations. Reduced row echelon form. Regular matrices: calculation of the inverse by the Gauss-Jordan method Basic problem solving Systems of linear equations. Equivalent systems. Rouché-Fröbenius theorem. Gauss-Jordan reduction. Solving systems of linear equations. Applications
Specific Objectives
Matrices are the fundamental tools we will work with and will then use in engineering applications. It is necessary to know in detail its basic properties. From the properties of the matrices and especially using the elementary row operations the solution of the systems of linear equations is considered. Numerical resolution algorithms are introduced.
Dedication
5h Large group + 5h Medium group + 14h Self StudyDefinition of a determinant. Fundamental properties. Determinant of a triangular matrix. Determinant of a block diagonal matrix. Calculation of determinants. Gauss method. Expression of the determinant. Development of a row and a column. Determinant of matrix multiplication. Determinants and matrix inversion. Cramer's rule.Geometric applications: Volume of a parallelepiped. Vector product. Properties. Examples of determinants calculated by reducing the matrix to triangular form. Exercises to determine if a matrix is invertible, and if so get its inverse. Solving systems of linear equations using Cramer's Rule.
Specific Objectives
Define the alternating multilinear forms, of which the determinant is a special case. From its definition shows some basic properties, all without having to explain the development of the determinant. Calculate the determinant of a matrix by applying numerical row elementary operations to reduce it to the triangular shape. After introducing the basic properties of permutations make explicit the crucial and its development. Using the properties of alternating multilinear forms of showing that the determinant of a matrix is the product of determinants. Cofactor matrix is defined and used in the calculation of the inverse matrix. Practicing proper elementary row operations and become aware of the possible numerical programming method. Working this second method to invert a matrix, as already know to transform it into reduced row echelon form by row.
Dedication
4h Large group + 4h Medium group + 11h 12m Self StudyDefinitions and examples. Image and Kernel subspaces. Monomorphism, epimorphism, isomorphism. Basic properties. Vector space of linear maps. Maps over and onto finite dimensional vector spaces. Associated matrix. Computation of basis for the image and the kernel of a linear map. Fundamental theorem. Composition of linear maps. Ring of endomorphisms. Inverse of an isomorphism. Matrix associated with a composition.. Basis change in a vector space. Connection of base change with the matrix associated with a linear map. The aim is to solve some problems on linear maps defined over infinite dimensional vector spaces, although focus shall be primarily on the finite dimensional case. Using the associated matrix and the theory of vector spaces and linear systems, basis for the kernel and image shall be obtained. Problems on the composition of linear maps. Computation of the matrix associated with the inverse map of an isomorphism.
Specific Objectives
Introduce the fundamental concepts and relate them to the mathematical contents covered in other courses. Acquaint the student with vectors other than the familiar examples deriving from physical applications. Relate the injectivity and surjectivity of a linear map to its kernel and image. Justify the correspondence between invertible and bijective linear maps. Relate the composition of linear maps to the product of their associated matrices. Computation of basis and analysis of their properties. Highlight the theory of base change, which shall be paramount in subsequent units: pay special attention to the relation between vector components in different basis and base vectors. Relate new concepts to the student's general background: vector spaces, matrix properties, resolution of linear systems, rang of a vector system and implicit equations of a subspace. Relate the contents of this topic to the properties they already know about matrices. . Insist on geometric applications.
Dedication
3h Large group + 4h Medium group + 9h 48m Self StudyBilinear forms. Examples and basic properties. Matrix associated to a bilinear form. Changing the base. Symmetric bilinear form. Quadratic form. Definite forms. Canonical form and normal form of a real symmetric bilinear form. Problems are solved by reduction of a symmetric bilinear form to its canonical form and normal. Change of basis. Definition of inner product. Examples. Basic properties. Orthogonal subspace. Orthogonal and orthonormal basis. Orthogonal projection. Pythagoras theorem and law of the parallelepiped.Fourier coefficient. Schwarz inequality, Bessel and triangular. Method of Gram-Schmidt. Geometric interpretations. Properties of inner product. Orthogonal projection onto a subspace. Geometric interpretations.
Specific Objectives
To develop the properties of bilinear forms, especially the symmetrical, preparing its subsequent application in Calculus. Working properties of symmetric bilinear forms using the method of elementary row operations they already know. Present the definitions and general properties and continually interpret in real Euclidean space of three dimensions with which the student is familiar. It aims at an abstract knowledge of Euclidean space applications are working especially geometry. Acquiring skill in use of abstract properties. General properties apply to real three-dimensional Euclidean space.
Dedication
4h Large group + 4h Medium group + 11h 12m Self StudyEigenvalues and eigenvectors. Characteristic polynomial. Diagonalization general theorem. Diagonalization problems. Elementary diagonalization theorem. Examples.Trigonalization basic theorem. Examples. Cayley-Hamilton. Examples and applications. Triangulation problems.
Specific Objectives
Being able to diagonalize a matrix facilitates the obtaining of its basic properties and its manipulation. This will be crucial in applications.
Dedication
4h Large group + 4h Medium group + 11h 12m Self StudyDefinitions and basic properties. Transposed. Associated matrices. Normal operators. Properties. Spectral theorem. Problems of normal operators and properties of normal matrices. Real normal operators. Symmetric and orthogonal operators. Spectral theorem for symmetric operators.Geometric interpretations. Real normal operator problems.
Specific Objectives
Most matrices that appear in engineering are symmetric matrices. It should be clear that these are diagonally orthogonal. Geometric applications of orthogonal matrices are also essential.
Dedication
4h Large group + 4h Medium group + 11h 12m Self Study