Mathematical Analysis & Linear Algebra
Course Description:
Mathematical Analysis: Mathematical Induction. Real numbers, sequences of real numbers, sequential completeness. Limit of a sequence, convergence tests. Series of real numbers, convergence tests. Calculus of functions of one variable. Trigonometric and inverse trigonometric functions. The notions of limit and continuity of a function, fundamental theorems. Derivative of a function, basic theorems, Taylor-McLaurin formula. Power series (Taylor – Mac-Laurin). The indefinite integral, basic methods of integration: integration by parts, the substitution method, integration of rational functions, trigonometric integrals. The Riemann integral of a real function, definition, examples, properties and applications. Improper integrals of first and second type: definition, simple and absolute convergence. Calculation and convergence tests. The integral test for convergence of series.
Linear Algebra: Vector calculus, vector products. The equations of the line and the plane in 3-space and applications. The sphere, cylindric and conic surfaces. Surfaces of 2nd degree, projection of a space curve on the coordinate planes. Matrices, determinants, rank of a matrix. Linear systems of equations, Gauss elimination method, the method of Cramer, invertible matrices. Vector spaces and subspaces. Linear span, linear dependence-independence, basis of a vector space, change of basis matrix. Linear functions (definition, kernel, image, matrix). Linear transformations, examples. Eigenvalues and eigenvectors of linear transformations and matrices Cayley-Hamilton theorem, matrix diagonalization. Orthogonal and symmetric matrices. Quadratic forms and applications.
- Semester 1
- Teaching hours 6
- Website https://helios.ntua.gr/course/view.php?id=1627
- Instructors I. Tsinias (Coordinator) S. Lampropoulou
Prerequisite Knowledge
Elementary school knowledge of Algebra, Analytic Geometry, Trigonometry and Calculus.Course Units
| # | Title | Description | Hours |
|---|---|---|---|
| 1 | Real numbers, Sequences, Convergence, | Mathematical Induction. Real numbers sequences of real numbers, sequential completeness. Limit of a sequence, convergence tests. | 3Χ3=9 |
| 2 | Series, Convergence tests. | Series of real numbers, convergence, absolute convergence, Series with nonnegative terms, Alternating series, convergence tests. | 3Χ3=9 |
| 3 | Differential Calculus | Calculus of functions of one variable. Trigonometric and inverse trigonometric functions. The notions of limit and continuity of a function. Derivative of a function, fundamental theorems. Taylor- Maclaurin formula. Power series (Taylor – Mac-Laurin). | 3Χ3=9 |
| 4 | Integral Calculus | Indefinite integral, methods of integration. The Riemann integral (definition, basic properties). Applications of definite integral in the calculation of the area, the arc length and the volume of a solid of revolution. | 3Χ3=9 |
| 5 | Improper Integrals | Improper integrals of first and second type, calculation and convergence tests. The integral test for convergence of series. | 1Χ3=3 |
| 6 | Complex numbers, Vector Calculus | The complex numbers. Introduction to vectors, vector products. | 2Χ3=6 |
| 7 | The Line in space, the Plane, Surfaces | The line in 3-space and applications. The plane and applications. The sphere, cylindric and conic surfaces. Surfaces of 2nd degree, projection of a space curve on the coordinate planes. | 3Χ3=9 |
| 8 | Matrices, determinants, linear systems | Introduction to matrices. Determinants, rank of a matrix. Linear systems, Gauss elimination method, the method of Cramer, application to inversion of matrices. | 3Χ2=6 |
| 9 | Vector spaces, Basis, Dimension | Vector spaces and subspaces. Linear span, linear dependence-independence, basis of a vector space, change of basis matrix. | 2Χ3=6 |
| 10 | Linear Functions | Linear functions (definition, kernel, image, matrix). Linear transformations, examples. | 3Χ2=6 |
| 11 | Eigenvalues and Eigenvectors, Quadratic Forms | Eigenvalues and eigenvectors of linear transformations and matrices (characteristic polynomial, Cayley-Hamilton theorem, matrix diagonalization). Orthogonal and symmetric matrices. Quadratic forms and applications | 3Χ2=6 |
Learning Objectives
Upon successful completion of the course, students will be able to know:
basic concepts and results of the differential and integral calculus of functions of a variable,
basic concepts and results of Linear Algebra and Vector Analysis.
Teaching Methods
| Teaching methods | Lectures, exercises, tests and homework |
|---|---|
| Teaching media | Blackboard presentations |
| Problems - Applications | Yes |
| Assignments (projects, reports) | Yes |
Student Assessment
- Final written exam: 70%
- Mid-term exam: 20%
- Assignments (projects, reports): 10%
Textbooks - Bibliography
- Κραββαρίτης Δ. (2018). Μαθήματα Ανάλυσης και Γραμμικής Άλγεβρας, Εκδ. Τσότρα. https://service.eudoxus.gr/search/#a/id:77111994/0
- Παντελίδης, Γ.(2008). Ανάλυση Τόμος Ι, Εκδ. Ζήτη, https://service.eudoxus.gr/search/#a/id:10966/0
- Ρασσιάς, Θ. (2017). Μαθηματική Ανάλυση Ι, Εκδ. Τσότρα, https://service.eudoxus.gr/search/#a/id:41955063/0
- Τσεκρέκος, Π. (2008). Μαθηματική Ανάλυση Ι. Εκδ. Μ. Αθανασοπούλου, Σ. Αθανασόπουλος Ο.Ε. https://service.eudoxus.gr/search/#a/id:45389/0
- Καδιανάκης Ν. Καρανάσιος Σ. (2011). Γραμμική Άλγεβρα, Αναλυτική Γεωμετρία & Εφαρμογές. Εκδ. Τσότρα. https://service.eudoxus.gr/search/#a/id:68382505/0
- Φελλούρης, Α. (2017). Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία. Εκδ. Τσότρα, https://service.eudoxus.gr/search/#a/id:68382520/0
- Παντελίδης, Γ., Κραββαρίτης, Δ., Νασόπουλος, Β. & Τσεκρέκος, Π.(2015). Εκδ.Τσότρα, https://service.eudoxus.gr/search/#a/id:59364446/0
Lecture Time - Place:
- Monday, 08:45 – 10:30,
Rooms:- Γενικές Έδρες ΑΜΦ 4
- Friday, 12:45 – 13:30,
Rooms:- Γενικές Έδρες ΑΜΦ 4