## 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
 Prof. MARCUS Andrei, Ph.D.,  marcusmath.ubbcluj.roLect. PELEA Cosmin Razvan, Ph.D.,  cpeleamath.ubbcluj.roLect. SACAREA Cristian, Ph.D.,  csacareamath.ubbcluj.ro
 Aims Notions and results from linear algebra. Applications
 Content Chapter I. VECTOR SPACES 1. Vector spaces and linear maps. Subspaces. Examples. 2. Linear dependence and independence. 3. The Steinitz exchange theorem. 4. Dimension. Dimension formulas. 5. The universal property of vector spaces. 6. The matrix of a linear map. Properties. 7. Change of base. Chapter II. SYSTEMS OF LINEAR EQUATIONS. 1. Sistems of linear equations. 2. Determinants; the inductive definition. 3. The rank of a matrix. 4. Invertible matrices. 5. Solving linear systems. The theorems of Cramer, Kronecker-Capelli and Rouche. 6. Algorithmic methods in linear algebra. Chapter III. EIGENVALUES AND EIGENVECTORS. 1. Eigenvectors and eigenvalues. The Hamilton-Cayley theorem. 2. Eigenspaces. Diagonalizable and triangularizable matrices. 3. The Jordan canonical form. Chapter IV. BILINEAR AND QUADRATIC FORMS. 1. Bilinear and quadratic forms. 2. Positively (semi)definite forms. 3. Sylvester's law of inertia. 4. Inner product. Unitary and Euclidean spaces. Gram-Schmid orthogonalization. 5. The adjoint of a linear operator. 6. Unitary, hermitian and anti-hermitian matrices. 7. Normal matrices and spectral theorems.
 References 1. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003. 2. G. CALUGAREANU, Lectii de algebra liniara, Litografiat Univ. Babes-Bolyai, 1995. 3. I.D. ION, N. RADU, Algebra (ed.4), Editura Didactica si Pedagogica, 1990. 4. N. BOURBAKI, Algebre, chap. 1-3, Ed. Hermann, Paris 1970. 5. I.V. PROSKURIAKOV: Problems in linear algebra, Mir Publishers, Moscow 1978. 6. S. CRIVEI: Basic Abstract Algebra, Casa Cartii de Stiinta, Cluj-Napoca 2002. 7. A. MARCUS: Algebra [http://math.ubbcluj.ro/~marcus] 8. S. AXLER: Linear algebra done right. Springer-Verlag, New York, 1997. 9. P. GABRIEL: Matrizen, Geometrie, Lineare Algebra, Birkhauser-Verlag, Basel-Boston-Berlin 1996. 10. I. PURDEA, C. PELEA, Probleme de algebra, EFES Cluj-Napoca 2005.
 Assessment Homeworks (20%). Exam. (80%)
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject