Description
The purpose of the course “Mathematical Analysis and Linear Algebra”, is to provide the students with the essential mathematical knowledge, skills and techniques, in order to be able to handle a wide range of problems occurring in the field of computer science.
Content
The outline of the course is as follows:
Linear Algebra
Matrices: Special Forms of Matrices, Matrix Operation and Properties, Matrix Inversion, Elementary Row Operations, Gaussian Elimination, Inverse Matrix Computation. Determinants: Computation Methods, Laplace’s expansion formula – Properties, Matrix Inversion Formula using determinants.
Linear Systems: Linear Systems as Matrix Equations, Square and Rectangular Systems, Solution using Matrix Inversion, Cramer’s Method, Augmented Matrix Method, Parametric Systems.
Sequences – Series
Sequences: Limit and Convergence, Computation of Limits, Indeterminate Forms.
Series Convergence Criteria: Comparison Criterion, D’Alembert Criterion, Cauchy Criterion, Ratio Criterion, Absolute Convergence Criterion, Leibniz’s Criterion.
Power Series: Radius of Convergence. Interval of Convergence.
Differential Calculus
Real Functions, Function Limits, Continuity, Derivatives, Derivative Properties and Applications, Mean Value Theorem, Taylor’s Theorem – Series, De Hospital’s Rule, Study of a Function.
Integral Calculus
Indefinite Integral: Linearity of the Integral, Integration by Parts, Change of Variable.
Definite Integral: Basic Properties, Fundamental Theorems of Calculus, Geometric Applications.
Structure
The course is structured as follows:
Teaching: 5hrs/week theory
Total workload per semester (13 weeks):
Lectures: 5 x 13 = 65 hrs
Homework: 18 x 6 = 108 hrs
Collaboration/Communication: 7 hrs
Total: 180 hrs
Evaluation
The final exam of the course is comprised of 5-6 problems involving the following topics:
Matrix – Determinants operations. Solution and study of parametric linear systems of equations
Determination of convergence/divergence of series – Computation of the interval of convergence of power series
Study of the features of a real function
Applications of differential calculus theorems (Mean value theorem, Taylor’s theorem, De Hospital’s rule, etc).
Computation of definite and indefinite integrals
Geometric application of definite integrals.
The above evaluation scheme is announced to the students (a) through the course’s homepage, (b) during the first lectures in the class.
Oral exams
The purpose of the course “Mathematical Analysis and Linear Algebra”, is to provide the students with the essential mathematical knowledge, skills and techniques, in order to be able to handle a wide range of problems occurring in the field of computer science. Having successfully completed the course the students is expected to be able to:
Recognize the basic matrix categories and be able to execute matrix operations. He or she should also be able to evaluate whether two matrices have the appropriate dimensions to participate a matrix operation and particularly to recognize that matrix multiplication is not commutative.
Calculate the row echelon form of given matrix, using elementary row operations through the Gauss – Jordan elimination Algorithm.
Calculate the determinant of a square matrix, using the Laplace expansion formula. He or she should also be able to recognize the high complexity of this type of computation in large matrices and compare it with the efficiency of other techniques (matrix triangularization).
Solve linear systems of equations by selecting the appropriate matric technique. Furthermore, he or she should be able to study linear systems whose coefficients are dependent uppon a real parameter.
Understand the concept of the limit of sequences and be able to compute such limits using the corresponding rules and techniques.
Understand the concept of series, as the sum of infinite number of terms of a sequence, while being able to recognize the difference between convergent and non-convergent series. Furthermore, he or she, should be able to distinguish series with positive terms and apply the corresponding convergence criteria. Finally, the student should be able to understand the difference between power series and generic series and be able to compute the convergence interval of the former.
Recall the basic concepts about real functions (limit, continuity, derivatives) and be able to calculate derivatives using the rules of differentiation.
Apply theorems and results from the theory of differential calculus such as the mean value theorem, De Hospital’s rule and Taylor’s theorem.
Apply the procedures required for the study of a real function and draw its plot.
Calculate indefinite integrals, applying one of the three main integration techniques (linearity, integration by parts, change of variable), while being able to predict the one that is likely to produce the result.
Understand the difference between definite and indefinite integrals and be able to apply fundamental results from the theory of integral calculus, to obtain the computation of certain types of integrals. Apply the techniques of definite integrals for the computation of geometric quantities (area, volume) as required.
1. Stewart J., Single Variable Calculus, 7th edition, ISBN: 0534218288, Brooks/Cole Publishing Co, February 7, 2011.
2. Thomas George B., Weir Maurice D., Hass Joel R., Thomas’ Calculus, 12th Edition, 2009, ISBN: 0321587995, Addison Wesley.
3. Strang G., Linear Algebra and its applications, 2009, Thomson, Brooks/Cole.
4. Apostol T.M., Calculus. Volume I. One variable Calculus, with an introduction to Linear Algebra, Second Edition, John Wiley and Sons, Inc., New York, 1991, ISBN-13: 978-0471000051.
