Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate

SUBJECT

Code
Subject
MML0001 Algebra 1 (Linear Algebra)
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
1
2+2+0
fundamental
compulsory
Mathematics and Computer Science
1
2+2+0
fundamental
compulsory
Applied Mathematics
1
2+2+0
fundamental
compulsory
Teaching Staff in Charge
Assoc.Prof. PELEA Cosmin Razvan, Ph.D.,  cpeleamath.ubbcluj.ro
Prof. MARCUS Andrei, Ph.D.,  marcusmath.ubbcluj.ro
Lect. MODOI Gheorghe Ciprian, Ph.D.,  cmodoimath.ubbcluj.ro
Aims
To introduce basics of linear algebra. To present basic properties of matrices, operations and applications of matrices. To solve systems of linear equations. To determine the eigenvalues and the eigenvectors of a matrix, to study the diagonalizability of a matrix. To determine the canonical form of a quadratic form.
Content
1. Preliminaries: groups, rings and fields.
2. Vector spaces. Examples.
3. Subspaces. Examples.
4. Linear maps
5. Linear independence. Basis. The universal property of vector spaces. The Steinitz exchange theorem. Dimension. Dimension formulas.
6. The matrix of a linear map. Properties. Change of base.
7. Determinants. The rank of a matrix. Invertible matrices.
8. Solving linear systems. The theorems of Cramer, Kronecker-Capelli and Rouche. Algorithmic methods in linear algebra.
9. Eigenvectors and eigenvalues. Eigenspaces. Diagonalizable matrices. The Hamilton-Cayley theorem.
10. Bilinear and quadratic forms. The canonical form of a quadratic form.
References
1. S. AXLER: Linear algebra done right. Springer-Verlag, New York, 1997.
2. N. BOURBAKI, Algebre, chap. 1-3, Ed. Hermann, Paris, 1970.
3. G. CALUGAREANU, Lectii de algebra liniara, Litografiat Univ. Babes-Bolyai, 1995.
4. S. CRIVEI: Basic Abstract Algebra, Casa Cartii de Stiinta, Cluj-Napoca, 2002.
5. P. GABRIEL: Matrizen, Geometrie, Lineare Algebra, Birkhauser-Verlag, Basel-Boston-Berlin, 1996.
6. I.D. ION, N. RADU, Algebra (ed.4), Editura Didactica si Pedagogica, 1990.
7. A. MARCUS: Algebra [http://math.ubbcluj.ro/~marcus]
8. C. NASTASESCU, I. STANESCU, C. NITA, Matematica, Elemente de algebra superioara, Editura Didactica si Pedagogica, Bucuresti, 1995.
9. I. PURDEA, C. PELEA, Probleme de algebra, EIKON, Cluj-Napoca, 2008.
10. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003.
Assessment
Homeworks or tests (25%). Exam. (75%)
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject