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., cpelea@math.ubbcluj.ro
Prof. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.ro
Lect. MODOI Gheorghe Ciprian, Ph.D., cmodoi@math.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.      	
Bibliography	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%)