SWIM 2025
Linear Algebra
Based on Artin's Algebra, this course covers core concepts of linear algebra. It starts with Subspaces of $\textbf{R}^{n}$ and introduces the concept of Fields. The definition and properties of Vector Spaces are discussed, followed by Bases and Dimension, techniques for Computing with Bases, and Direct Sums. The course then moves to Linear Operators, the Dimension Formula, and representing linear maps using the Matrix of a Linear Transformation. Key topics like Eigenvectors and Eigenvalues, the Characteristic Polynomial, and finding Triangular and Diagonal Forms are also covered.
Lecture Schedule & Resources
Note: The schedule is tentative and may be subject to change during the program.
| Lecture | Topics Covered | Speaker | Resources |
|---|---|---|---|
| Lecture 1 May 7 (Wed) |
Introduces subspaces (definitions, examples in $\textbf{R}^{n}$) and fields (axioms, $\textbf{Q}/\textbf{R}/\textbf{C}$), linking algebraic structures to geometric intuition. | Krishna Hanumanthu – Gobinda Sau | |
| Lecture 2 May 9 (Fri) |
Generalizes subspaces to abstract vector spaces over arbitrary fields, emphasizing axioms, examples (matrices, function spaces), and scalar-field dependencies. | Krishna Hanumanthu – Gobinda Sau | |
| Lecture 3 May 12 (Mon) |
Defines bases, linear independence, and dimension as measures of a vector space’s "size." | Sruthymurali – Gobinda Sau | |
| Lecture 4 May 13 (Tue) |
Covers techniques for coordinate representation, basis transformations, and practical computations. | Sruthymurali – Gobinda Sau | |
| Lecture 5 May 16 (Fri) |
Decomposes vector spaces into subspaces, highlighting independence and complementary structures. | Sruthymurali – Gobinda Sau | |
| Lecture 6 May 21 (Wed) |
Introduces functions preserving vector operations (linear operators), with examples and properties. | A. K. Vijayarajan – Gobinda Sau | |
| Lecture 7 May 22 (Thu) |
Relates kernel and image dimensions via the Rank-Nullity Theorem. | A. K. Vijayarajan – Gobinda Sau | |
| Lecture 8 May 23 (Fri) |
Introduction to matrix representations of linear maps relative to different bases. Exploring the effects of basis changes on matrix representations. | A. K. Vijayarajan – Gobinda Sau | |
| Lecture 9 May 27 (Tue) |
Introduction to eigenvectors and eigenvalues and their geometric significance. | Divakaran D. – Gobinda Sau | |
| Lecture 10 May 28 (Wed) |
Methods for computing eigenvalues and eigenvectors. Introduction to the characteristic polynomial. | Divakaran D. – Gobinda Sau | |
| Lecture 11 May 28 (Wed) |
Deriving eigenvalues from polynomial roots using the characteristic polynomial. Introduction to matrix triangularization. | Divakaran D. – Gobinda Sau | |
| Lecture 12 May 29 (Thu) |
Matrix diagonalization: conditions for diagonalizability and the utility of using an eigenbasis. | Divakaran D. – Gobinda Sau |