Course title Geometry of plane and space  – Introduction in algebra
Code M106
Status Lectures and auditory exercises.
Level Elective course
Year 1. Semester 1.
ECTS 5 ECTS credits
Lecturer Darija Marković, Associate Professor; Dr. Darija Brajković Zorić, Assistant
Course objective The objective of the course at the introductory level based on geometry of plane and space is to make students familiar with fundamentals of linear algebra.
Prerequisites None
Learning outcomes: After successfully completed course, students will be able to:

  1. know about term of vector and basic vector operations in plane and space with corresponding applications,
  2. understand concept of introduction of linear operator on vector space, as well as connection with the term of matrix, and know matrix calculation,
  3. generalize all the mentioned  terms into several dimensions and at more abstract level,
  4. adopt basic principles of proving of mathematical assertion.
Teaching activity ECTS Learning outcome Students activity Methods of evaluation Points
min max
Class attendance 1 20% Class attendance  Evidence list 0 10
Knowledge test (preliminary exam) 2 40% Preparation for written examination Written  preliminary exam 0 40
Final exam 2 40% Repetition of teaching materials. Oral exam (and written exam). 0 50
Total 5 100% 0 100
Consultations By appointment.
Gained competencies Knowledge on fundamentals of linear algebra based on geometry of plane and space (elementary vector operations in plane and space, symetric and orthogonal linear operators in plane and space, square matrices, curves of the second order).
Content (Course curriculum)
  1. Operations with vectors. Linear dependence and independence of vectors. Basis of vector spaces. Coordinate system. Norm of vectors. Distance between two points. Cauchy – Schwarz – Buniakowsky inequality. Vector dot/scalar product. Direction cosine. Projection of vector to the line and plane. Gramm – Schmidt orthonormalization process.
  2. Square matrix of the second and third order and their determinants. Orientation – right and left basis and coordinate systems. Vector cross product. Algebraic properties. of the vector product. Geometrical properties of the cross product. Scalar triple product. Vector triple product. Jacobi identity. Straight line and plane in space. Hesse normal form of line and plane.
  3. Linear operators in plane. Examples of operators: axial symmetry, central symmetry, homothety, orthogonal projection, rotation. Basic properties of the linear operator. Operations with linear operators – vector space . Products and power of the linear operator. Matrix of the linear operator. Algebra of the matrix of the second order. Contraction and dilatation of the plane – eigenvectors and eigenvalues of the linear operator. Symmetric linear operator in the plane. Orthogonal linear operator in the plane. Diagonalization of the symmetric linear operator. Quadratic forms. Curves of the second order.
  4. Linear operators in space . Examples. Transfer of all definitions from plane. Symmetric linear operator in the space. Surfaces of the second order.
Recommended reading
  1. R.Scitovski, Geometrija ravnine i prostora, reviewed course materials available on the course website, Department of Mathematics, University of Osijek, Sveučilište u Osijeku, 2011.
  2. S. Kurepa, Uvod u linearnu algebru, Školska knjiga, Zagreb, 1978.
Additional reading
  1. D.Bakić, Linearna algebra, Školska knjiga, Zagreb, 2008.
  2. N. Elezović, Linearna algebra, Element, Zagreb, 2001.
  3. J.Hefferon, Linear Algebra, Saint Michael’s College, Colchester, Vermont, USA, 2011 – freely available at:
  4. D.Jukić, R.Scitovski, Matematika I, Department of Mathematics, University of Osijek, Sveučilište u Osijeku, Osijek, 2004.
Instructional methods Lectures and auditory exercises are obligatory to all students.
Exam formats The exam is taken after completion of lectures and exercises, and it consists of a written and an oral part. There are 2 midterm exams during the semester that cover the entire course syllabus. Once a student has successfully passed all mid-term exams, he/she does not have to take the written part of the exam.
Language Croatian
Quality control and successfulness follow up Conducting an anonymous survey among students.
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