This course develops linear algebra through a simplified approach emphasising applications to machine learning and statistics centered subjects via the cross-filling method (rank-one decomposition) that simplifies traditional Gaussian elimination with boarder applications to other application areas.

Covered topics include:
• Matrix computation strategies: Cross-filling, sum ↔ product equivalence
• Vector spaces and subspaces: Null space, column space, the four fundamental subspaces
• Projection theory: Oblique and orthogonal projections, compatible families, spectral decompositions
• Determinants: Axiomatic properties, Laplace expansion, Cayley-Hamilton theorem
• Eigenvalues and spectral decomposition: Lagrange interpolation approach to diagonalization
• Normal matrices and applications: Hermitian/unitary matrices, positive definiteness, singular value decomposition (SVD)

Applications include:
• Multi-variable Linear Regression and Data Analysis
• Differential Equations

Pedagogical note: This curriculum uses cross-filling and Lagrange interpolation as primary computational methods, offering alternative and often more elegant techniques than traditional approaches.

None

The textbook is a good reference to this lecture. We will mainly follow my lecture notes available at https://math-postech.github.io/courses/MATH203/2026-spring/

The material refers to the website https://math-postech.github.io/courses/MATH203/2026-spring
The plan might subject to change.

The lecture at 2026-03-04 and 2026-05-27 will be held online at https://utoronto.zoom.us/j/2195915645

• Week 01 (02-23 -- 02-26): Ch1 §1.1-1.2 — Matrix Arithmetic
• Week 02 (03-02 -- 03-05): Ch1 §1.3 — Row reduction, Cross-Filling method
• Week 03 (03-09 -- 03-12): Ch1 §1.3-1.4, Ch2 §2.1 — Solving Systems, Subspaces
• Week 04 (03-16 -- 03-19): Ch2 §2.1-2.2 — Vector Spaces
• Week 05 (03-23 -- 03-26): Ch2 §2.3-2.4 — Four Subspaces, Basis & Dimension
• Week 06 (03-30 -- 04-02): Ch2 §2.4-2.5, Ch3 §3.1 — Full Rank & Projections
• Week 07 (04-06 -- 04-09): Ch3 §3.2-3.4 — Projection Theory + Midterm Review
• Week 08 (04-13 -- 04-16): MIDTERM EXAM (Mon 04-13, 18:00-22:00) 3 hour exam inside this time window. Please refer to announcement.
• Week 09 (04-20 -- 04-23): Ch3 §3.5-3.7 — Constructing Projections
• Week 10 (04-27 -- 04-30): Ch4 §4.1-4.3 — Determinants
• Week 11 (05-04 -- 05-07): Ch4 §4.4-4.5 — Det Properties & Cayley-Hamilton
• Week 12 (05-11 -- 05-14): Ch5 §5.1-5.2 — Eigenvalues & Lagrange Interpolation
• Week 13 (05-18 -- 05-21): Ch5 §5.3-5.4 — Spectral Decomposition
• Week 14 (05-25 -- 05-28): Ch5 §5.5-5.6 — Diagonalization & Applications (05-25 Holiday)
• Week 15 (06-01 -- 06-04): Ch6 §6.1-6.5 — Normal Matrices, PD, SVD + Final Review
• Week 16 (06-08 -- 06-11): FINAL EXAM (Mon 06-08, 18:00-22:00) 3 hour exam inside this time window. Please refer to announcement.

Teaching Method:
This course teaches in person.

Homework:
4 assignments distributed throughout the semester:
• HW1 (Week 3): Ch1 — Matrix computation, cross-filling, solving linear systems
• HW2 (Week 6): Ch2-3 — Subspaces, basis & dimension, projections
• HW3 (Week 10): Ch4 — Determinants, Cayley-Hamilton theorem
• HW4 (Week 13): Ch5-6 — Spectral decomposition, normal matrices, SVD

Homework problems are provided as separate PDF files.

Tutorial Sessions:
• Time: THU 18:00-18:50 (all sections). The tutorial only starts from the 2nd week.
• Locations (by section):
- Section 01 (Li Qirui): MathBldg [206], [402], [104]
- Section 02 (Kim Younjin): Science BldgⅢ [109], [111], [113]
- Section 03 (Yu Seungook): Hogil Kim Building [306], [307], [308]
• Purpose: Reinforce lecture content, practice cross-filling and computational methods

Coordination:
This course has 3 sections taught by different instructors (Li Qirui, Kim Younjin, Yu Seungook). All sections share the same syllabus and exam schedule.