Outline [28 lectures with 4 optional + 1 slack; 2 need to be skipped to make room for tests]:
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