Linear Algebra, Math 405 Fall 2008

Course Information

• PREREQUISITES: MATH 240  or MATH 461
  
   

• Course Description: This course contains an abstract treatment of finite dimensional vector spaces, linear transformations and their invariants.
  
  
• Contents
  
  Vector spaces
      Vector spaces and subspaces
      Linear combinations, span and linear (in-)dependence
  
      Note: On the transitivity of linear spans [ pdf file ]
  

       Assignment #1 [ pdf file ] ... with answers [ pdf file]  

      Bases and dimension
      Row equivalence in R^N and rank
      Coordinates
  
      Note: An exchange theorem and the notion of dimension [ pdf file ]
  

       Assignment #2 [ pdf file ] ... with answers [ pdf file]  

  Linear transformations
      Linear transformations (homomorphism, isomorphism, ...)
      Vector spaces of finite and infinite type; Rank and nullity, Sylvester theorem
      (*) Direct sums and products of vector spaces
  

       Assignment #3 [ pdf file ] ... with answers [ pdf file]  

      Matrix representation of linear transformations
      Sum and product of matrices; change of bases
  
      Note: On matrix representation of linear transformations [ pdf file ]
  

       Assignment #4 [ pdf file ] ... with answers [ pdf file]  

      Linear functionals and duality
  

       Assignment #5 [ pdf file ] ... with answers [ pdf file]  
  
       Midterm [ pdf file ] ... with answers [ pdf file]  

  Solution of linear systems of equations
      Homogeneous and inhomogeneous systems
      Determinants: definition and properties; Permutations and evaluation
      Elementary matrix operations
      Invertible matrices; Cramer's rule
  

       Assignment #6 [ pdf file ] ... with answers [ pdf file]  

  Inner Product Spaces
      Inner product
      Orthonormal Bases; Gram-Schmidt; direct decompositions
      Bessel, Parseval and Cauchy-Schwartz (in-)equalities
  

       Assignment #7  [ pdf file ] ... with answers [ pdf file]  

      Adjoints
      Positive, unitary and normal operators
      The spectral theorem
  
      Note:  On the the orthogonal decomposition of Hilbert spaces [ pdf file ]

       Assignment #8 [ pdf file ] ... with answers [ pdf file]  

  Invariant decompositions
      Direct sum decomposition's
      Invariant subspaces
      Eigenvectors and eigenvalues
  

       Assignment #9 [ pdf file ] ... with answers [ pdf file]  

      Polynomials: the algorithm of Euclid; Factorization and (multiple) roots
      Characteristic polynomial
      Triangulation of matrices
      Diagonalizable operators
      Jordan canonical forms
  

       Assignment #10 [ pdf file ] ... with answers [ pdf file]  

      Cayley-Hamilton: characteristic and minimal polynomials; companion matrix
      Applications
  
      Note:  Normal matrices are unitarily diagonalizable [ pdf file ]

       Assignment #11 [ pdf file ] ... with answers [ pdf file]  

  Bilinear forms
      General forms
      Symmetric forms
      Sylvester theorem
      Principle coordinates
  
      Note:  Bilinear forms, quadratic forms and congruence [ pdf file ]

       Assignment #12 [ pdf file ] ... with answers [ pdf file]  

       Final ... [ pdf file ] ... with answers [ pdf file]  

  Euclidean geometry
      3D vectors; scalar and vector products
      Lines and relation between lines
      Planes: definition and characterizations
      Planes and lines
      Change of coordinates; translation, rotation and reflection
  
  

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  Epilogue - The sin(x) subroutine in MATLAB

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  References (Partial List)

• Linear Algebra, 2nd Edition, by Hoffman & Kunze,  Published by Prentice Hall. ISBN: 0135367972
• Linear Algebra, 3rd Edition, by S. Lang,  Published by Springer-Verlag.  ISBN: 0387964126
• Linear Algebra in Action by Harry Dym,  Graduate Studies in Math v. 78, American Math Society, 2007
  
• Linear Algebra, by Peter Lax
  
• Applied Linear Algebra, by Gilbert Strang
• Lecture Notes