# 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.

### 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:

1. basic concepts and results of the differential and integral calculus of functions of a variable,

2. basic concepts and results of Linear Algebra and Vector Analysis.

### Teaching Methods

 Teaching methods Lectures, exercises, tests and homework Blackboard presentations Yes Yes

### Student Assessment

• Final written exam: 70%
• Mid-term exam: 20%
• Assignments (projects, reports): 10%

### Textbooks - Bibliography

1. Κραββαρίτης Δ. (2018). Μαθήματα Ανάλυσης και Γραμμικής Άλγεβρας, Εκδ. Τσότρα. https://service.eudoxus.gr/search/#a/id:77111994/0
2. Παντελίδης, Γ.(2008). Ανάλυση Τόμος Ι, Εκδ. Ζήτη, https://service.eudoxus.gr/search/#a/id:10966/0
3. Ρασσιάς, Θ. (2017). Μαθηματική Ανάλυση Ι, Εκδ. Τσότρα, https://service.eudoxus.gr/search/#a/id:41955063/0
4. Τσεκρέκος, Π. (2008). Μαθηματική Ανάλυση Ι. Εκδ. Μ. Αθανασοπούλου, Σ. Αθανασόπουλος Ο.Ε. https://service.eudoxus.gr/search/#a/id:45389/0
5. Καδιανάκης Ν. Καρανάσιος Σ. (2011). Γραμμική Άλγεβρα, Αναλυτική Γεωμετρία & Εφαρμογές. Εκδ. Τσότρα. https://service.eudoxus.gr/search/#a/id:68382505/0
6. Φελλούρης, Α. (2017). Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία. Εκδ. Τσότρα, https://service.eudoxus.gr/search/#a/id:68382520/0
7. 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

© 2017 School of Civil Engineering, ΕΜΠ