## 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

**Jaroslaw (Jarek) Kwapisz**

Professor of Mathematics

Department of Mathematical Sciences

Montana State University

P.O. Box 172400

Bozeman, MT 59717-2400

Office: 2-240 Wilson Building

Phone: (406) 994 5353

jarek@math.montana.edu