Mathematical Analysis – Linear Algebra

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Learning Outcomes

After successfully completing this course, students are expected to have acquired the basic knowledge of Mathematical Analysis regarding sequences, series, functions of one variable (differentiation, integration) and the numerical methods that are necessary to Computer Science. After successfully completing this course, students are expected to be familiar with the notion of the algebraic structure, and to be equipped with the necessary knowledge regarding matrices and determinants, as well as with their use for the solution of linear systems. Students are also expected to be familiar with the basic knowledge regarding the inner product, the characteristics values and vectors of matrices and their use for the diagonalization, the orthogonal diagonalization and the evaluation of the powers of matrices. The course directly supports most of the curriculum subjects and lessons: Note that during the lesson, specific examples of application are discussed in some of the above curricular subjects such as applications in telecommunication systems, cryptography, digital services such as e- learning, e-health. In conclusion, the students through the course process develop mathematical thinking and analyze, adapt their acquired knowledge to apply them to a variety of subjects in the field of study or the field of study, as well as to acquire new knowledge. In addition, they learn to solve complex or new problems in their scientific field of study by developing integrated as well as creative or innovative solutions and approaches while supporting their solutions and views in a methodical and scientific way. Finally, they learn to analyze and critically and responsibly choose ideas and information about those elements that concern them.

Course Contents

  • Sequences. Series.
  • Functions of One Variable.
  • Differentiation.
  • Iterative Methods for Solving Equations.
  • Polynomial Approximation of Functions.
  • Numerical Differentiation.
  • Indefinite Integral. Differential Equations.
  • Definite Integral.
  • Algebraic Structures: Groups. Rings.
  • Fields. Vector Spaces.
  • Basic Linear Algebra: Matrices. Determinants. Linear Systems.
  • Characteristics of a Matrix: Eigenvalues. Eigenvectors. Diagonalization of Matrices. Bilinear Forms.
  • Inner Product: Orthonormalization. Gram-Schmidt Algorithm.
  • Linear Transformation Matrices: Change of bases matrix. Matrix of a linear transformation.
  • M.Filippakis, Applied Analysis and Linear Algebra, Tsotras Publications, Athens 2017, 2nd Edition.
  • Χ. Moisiadis, Advanced Mathematics.
  • Teaching Notes.