線性代數

This course is supposed to equip students with basic notions and concepts of linear algebra needed for performing scientific computations. The main objective of the class is to make students familiar with the structure of algebraic objects (vectors, matrices, vector spaces, bases, decompositions) to the level necessary for performing real calculations and solving real problems. This practical approach will be combined with formal mathematical character of exposition.

undergraduate mathematics (preferably)

無備註

materials for homework will be published on e3 homepage

The total score is composed of 4 parts: 1. midterm 25% 2. final 25% 3. every class starts with a short quiz, ten best quizzes count for 25% of the final score 4. there will be 6-7 homework problem sheets, which will be solved during the class by randomly selected students; average of the performance at the blackboard counts for 25% of the final score

• Complex numbers

  folk, formal, and geometric definitions, operations on complex numbers

• Vectors

  definition, operations on vectors

• Matrices

  definitions, basic operations on matrices, high-level operations on matrices, functions of matrices

• Vector spaces

  definition, linear independence, basis, change of basis, orthonormalization process, dual basis, rank of matrices, tensor spaces

• Linear equations

  basic methods of solutions, BLAS and LAPACK

• Linear operators

  matrix representation, properties of linear operators, change of basis, functions of linear operators

• Eigenvalues and eigenvectors of linear operators

  definitions, geometric interpretation, characteristic polynomial, deficiencies, Jordan form of linear operators, eigendecomposition

• Eigenvalue and generalized eigenvalue problem

  methods of determination of eigenvalues and eigenvectors, diagonalization, LAPACK

• Singular value decomposition

  definitions, interpretation of singular values, generalized inverse

• Practical approach to problems in huge vector spaces

  methods of solving of huge linear equations (DIIS algorithm), methods of solving huge eigenvalue problems (Davidson method and related techniques)

• Elements of analytic geometry

  coordinates, area and volume of geometric objects, crossing points for geometrical objects, translations and rotations of geometric objects

for basic concepts: any of numerous textbooks for undergraduate algebra for higher level concepts: 1. D. S. Bernstein "Matrix mathematics", Princeton University Press, second edition, 2009 2. G. H. Golub and C. F. Van Loan "Matrix computations", Johns Hopkins University Press, third edition, 1996 3. J. H. Wilkinson "The algebraic eigenvalue problem", Oxford Science Publications, 1965 (reprinted 2004) for problems: S. Lipschutz "Schaum's Outline of Linear Algebra", McGrawHill, fifth edition, 2012

地點

SB202

時間

Thursday EFG

聯絡方式

by email: hwitek@mail.nctu.edu.tw