# Mathematical Analysis & Linear Algebra

## Course Description:

Mathematical Analysis: Real numbers (topology of R, supremum and infimum of a set, the Bolzano-Weierstrass theorem). Sequences of real numbers, convergence tests. Series of real numbers, convergence tests. Calculus of functions of one variable, fundamental theorems, Taylor-McLaurin formula, extremals. Power series (Taylor – Mac-Laurin). The indefinite integral, methods of integration. The Riemann integral (definition, tests of integrability, applications). Improper integrals of first and second type, calculation and convergence tests. The integral test for convergence of series.

Linear Algebra: The complex numbers. Vector calculus, the equations of line and 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 (characteristic polynomial, 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, Real numbers, Topology of R, supremum and infimum, Bozano-Weierstrass theorem, Sequences of real numbers, convergence tests. 2Χ3=6
2 Series, Convergence tests. Series of real numbers, convergence, absolute convergence, Series with nonnegative terms, Alternating series, convergence tests. 3Χ3=9
3 Differential Calculus Differential calculus of one variable, fundamental theorems, Taylor- Maclaurin formula, Exremals, Power series (Taylor – Mac-Laurin). 2Χ3=6
4 Integral Calculus Indefinite integral, methods of integration. The Riemann integral (definition, tests of integrability, applications). 4Χ3=12
5 Improper Integrals Improper integrals of first and second type, calculation and convergence tests. The integral test for convergence of series. 2Χ3=6
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

Με την επιτυχή ολοκλήρωση του μαθήματος, οι φοιτητές θα είναι σε θέση να γνωρίζουν

1. βασικές έννοιες και αποτελέσματα του διαφορικού και ολοκληρωτικού λογισμού συναρτήσεων μιάς μεταβλητής,

2. βασικές έννοιες και αποτελέσματα της Γραμμικής Αλγεβρας και Διανυσματικής Ανάλυσης .

### 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. Ρασσιάς, Θ. (2014). Μαθηματική Ανάλυση Ι, Εκδ. Τσότρα, https://service.eudoxus.gr/search/#a/id:41955063/0
2. Παντελίδης, Γ.(2008). Ανάλυση Τόμος Ι, Εκδ. Ζήτη, https://service.eudoxus.gr/search/#a/id:10966/0
3. Τσεκρέκος, Π. (2008). Μαθηματική Ανάλυση Ι. Εκδ. Μ. Αθανασοπούλου, Σ. Αθανασόπουλος Ο.Ε. https://service.eudoxus.gr/search/#a/id:45389/0
4. Φελλούρης, Α.(2009). Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία. Εκδ. Α.Φελλούρης, https://service.eudoxus.gr/search/#a/id:7041/0
5. Καρανάσιος, Σ. & Καδιανάκης, Ν.(2011). Γραμμική Άλγεβρα, Αναλυτική Γεωμετρία & Εφαρμογές. Εκδ. Καρανάσιος, Σ. & Καδιανάκης, https://service.eudoxus.gr/search/#a/id:6832/0
6. 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