# Linear Algebra

Course ID
1ΚΠ02
Επίπεδο
Είδος
Compulsory
Εξάμηνο
1
Περίοδος
Fall Semester
ECTS
5
Ώρες Θεωρίας
3
Ώρες Εργαστηρίου
1

#### Instructor

Aretaki Aikaterini

#### Assistant

Panagiotou Nikolaos

#### Description

• Matrix algebra-basic properties. Invertible matrices and their properties. • Determinants and their properties. Computing the inverse of a matrix. • Matrices and systems of linear equations. Rank of a matrix. Solving linear systems-Algorithms Gauss and Cramer. • Vector spaces and subspaces. Special vector subspaces (sum, intersection, orthogonal complement). Linear combinations, linear dependence-independence of vectors. Basis and dimension of a vector space. • Linear transformation – Kernel and Image of a linear transformation. Rank–nullity theorem. Matrix of a linear transformation. Similar matrices. • Inner product on a vector space. Norm of a vector. Orthonormalization of a basis. Gram-Schmidt algorithm. Orthogonal spaces and basic theorems. • Properties of eigenvalues and eingenvectors of a matrix. Theorem Cayley- Hamilton. Minimal polynomial. • Canonical forms. Diagonalizing of a matrix. Criteria diagonalizable matrix (linear transformation). Spectral theorem. Applications. • Quadratic forms-basic criteria for symmetric matrices. Applications to quadratic forms on the problems min-max of a function of several real variables.

#### Course objectives

Since the topic of the course is the study of fundamental concepts of the linear algebra such as vector spaces and linear transformation, the matrix theory is an essential tool in the study and modeling of many mathematical problems and the concept of the matrix is unfamiliar to the student, the main aim of the course is:

• to familiarize students with the concept of the matrix, to present the basic methods of solving linear systems, to study the basic and characteristic properties of the square matrices,
• the development of computational methods and techniques for the calculation of various sizes, and
• the theoretical study of properties of topics, such as vector spaces.

The course aims at the acquisition of knowledge and skills so that they can be applied to other courses of mathematics and computer science.

#### Textbooks/Bibliography

• Γραμμική Άλγεβρα-Θεωρία και Εφαρμογές, Γ. ΔΟΝΑΤΟΣ- Μ. ΑΔΑΜ, “Γ. ΔΑΡΔΑΝΟΣ – Κ. ΔΑΡΔΑΝΟΣ Ο.Ε.”, 1η έκδ./2008, ΑΘΗΝΑ,         31174
• Γραμμική Άλγεβρα με το Matlab: Νέα έκδοση, Στεφανίδης Γιώργος, ΜΑΡΚΟΥ ΚΑΙ ΣΙΑ Ε.Ε., 1η/2014, ΘΕΣ/ΝΙΚΗ, 41960366
• Εισαγωγή στη Γραμμική Άλγεβρα, Χατζάρας Ιωάνννης, Γραμμένος Θεοφ., ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 1η/2011, ΘΕΣ/ΝΙΚΗ, 18548920
• Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία,  Μυλωνάς Νίκος, ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 1η/2013, ΘΕΣ/ΝΙΚΗ, 32998759
• Γραμμική Άλγεβρα, Μάργαρης Αθανάσιος, ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 1η/2015, ΘΕΣ/ΝΙΚΗ, 50659814

#### Assessment method

Mandatory written exams at the end of the semester.