First, as a warmup, some fun applications meant to prompt a review of some fundamentals from Matrix Theory M 221

[3 lectures + 1 slack]

  • (to review Vectors) Modular Arithmetics and Error Detecting Codes (1.4)
  • (to review Matrix Multiplication) Hamming Code (Example 3.70, p241)
  • (to review Linear Systems) Finite Linear Games (Example 2.33, p109)

Now comes the first lump of the theoretical core of the course

[9 lectures]

  • Vector Spaces and Subspaces (6.1 and 2.3, 3.5) [2 lect]
  • Lin. Indep. , Basis, Dimension (6.2 and 3.5) [2 lect]
  • Change of Basis (6.3)
  • Linear Transformations (3.6 and 6.4) [2 lect]
  • Kernel and Range (6.5)
  • Matrix of Linear Transformation (6.6)

Two fun applications

[2 lectures with 1 optional]

  • Application: Tilings and Crystallographic Restriction
  • Application: Linear Codes as Subspaces (p 529) (Optional)

More theory focusing on symmetric matrices and orthogonality, theory interlaced with applications

[7 lectures with 1 optional]

  • Inner Product Spaces (7.1)
  • Gram-Schmidt Process and QR Decomposition (mention QR algorithm)
  • Orthogonal Diagonlization of Symmetric Matrices
  • Application: Dual Codes (Optional)
  • Normed Spaces and Some Useful Norms: Operator, Hamming, ...
  • Least Squares
  • Application: Reed Muller Code

Finally, theory and applications of linear transformations

[9 lectures with 1 optional]

  • Similarity and Diagonlization
  • Application: Linear Recurrence Relations (Th 4.38 p337)
  • Application (mention only? in lieu of 225): Systems of Linear Differential Equations
  • Application: Matrices, Graphs, Markov Chains
  • Perron Frobenius Theorem (with Proof)
  • Jordan Theorem (with proof only for a sharp class) [2 lectures]
  • Singular Value Decomposition
  • Application: Digital Image Compression