LINEAR  ALGEBRA
August -- December  2003

Lecture  1        (August  04, 2003)     Review of set theory
Lecture  2        (August  06, 2003)     Algebraic Preliminaries
Lecture  3        (August  08, 2003)     Vector Spaces 
Lecture  4        (August  11, 2003)     Subspaces 
Lecture  5        (August  13, 2003)     Generating systems 
Lecture  6        (August  18, 2003)     System of linear equations -- Gauss elimination
Lecture  7        (August  20, 2003)     Linear independence
Lecture  8        (August  22, 2003)     Maximal generating systems, Minimal linear independent subsets  -- Set theory  -- Zorn's lemma 
Lecture  9        (August  25, 2003)     Bases, Dimension of vector spaces  -- Set theory  -- cardinal numbers and their arithmetic 
Lecture  10      (August  27, 2003)     Dimension of vector spaces
Lecture  11      (August  29, 2003)     Affine Spaces -- Concept of Affine spaces, Affine subspaces  
Lecture  12      (September  01, 2003)     Affine Subpaces --  Continued  
Lecture  13      (September  03, 2003)     Linear maps  -- Group Homomorphisms, Linear Maps  
Lecture  14      (September  05, 2003)     Space of Linear Maps  
Lecture  15      (September  08, 2003)     Linear Maps and bases 
Lecture  16      (September  10, 2003)     Rank Theorem 
Lecture  17      (September  12, 2003)   Direct Sums and Projections, Dual Spaces 
*Lecture  18    (September  13, 2003)   Affine Maps 
Lecture  19      (September  15, 2003)   Dual Spaces 
Lecture  20      (September  17, 2003)   Quotient Spaces - - Isomorphism theorems, Exact Sequences 
Lecture  21      (September  19, 2003)   Group actions 
*Lecture  22    (September  20, 2003)   Affine Maps  --  Continued 
Lecture  23      (September  22, 2003)   Group actions  --  Continued 
Lecture  24      (September  24, 2003)   Matrices -- The Matrix of a Linear Map, Properties of matrices. Matrix algebra.  
Lecture  25      (September  26, 2003)   Matrices  --  Continued -- Transition matrices,  Change of basis --Similar matrices. 
Lecture  26      (October  01, 2003)   Matrices -- Continued  --  Matrix of a dual map
Lecture  27      (October  05, 2003)   Matrices -- Continued  -- Rank of matrices, Row and column ranks, Elementary operations 
Lecture  28      (October  06, 2003)   Matrices -- Continued  -- Triangular matrices, nilpotent operators  
Lecture  29      (October  09, 2003)   Matrices -- Continued  -- Elementary Matrices. 
Lecture  30      (October  10, 2003)   Matrices -- Continued  --  Elementary divisor theorem. 
Lecture  31      (October  13, 2003)   Applications of Elementary divisors theorem                                                                                               ---  Structure of finitely generated abelain groups,    Linear system of  Diophantine equations  
Lecture  32      (October  14, 2003)   Permutations
Lecture  33      (October  15, 2003)   Permutations --Continued 
Lecture  34      (October  17, 2003)   Multilinear and alternating maps
Lecture  35      (October  20, 2003)   Determinants  - -  Determinant functions
Lecture  36      (October  22, 2003)   Rules for computations -- Minor criterion for rank, Elementary operations
Lecture  37      (October  24, 2003)   Expansions of determinants, Cramer's Rule
Lecture  38      (October  27, 2003)   Orientations
Lecture  39      (October  29, 2003)   Volume and Determinants
Lecture  40      (October  31, 2003)   Eigen values, Spectral values, Minimal polynomials of linear operators
Lecture  41      (November 03, 2003)   Characteristic polynomial, Cayley-Hamilton theorem
Lecture  42      (November  05, 2003)   Prime divisors of Characteristic and Minimal polynomials
Lecture  43      (November  07, 2003)   Examples -- Eigenvalues, Spectral values,  Characteristic and Minimal polynomials. 
Lecture  44      (November  10, 2003)   Diagonalisable and triangulable operators
Lecture  45      (November  12, 2003)   Primary decomposition theorem
Lecture  46      (November  14, 2003)   Additive and multiplicative canonical decompositions
Lecture  47      (November  17, 2003)   Jordon canonical form over algebraically closed fields
Lecture  48      (November  19, 2003)   Jordon canonical form over arbitrary fields --- In particular over the field of real numbers
Lecture  49      (November  21, 2003)  
Lecture  50      (November  24, 2003)  

Back to the Home Page

Back to the Linear Algebra Course