Lecture notes
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Full lecture notes - working draft
| Date | Lecture Notes | Topics Covered |
|---|---|---|
| 2025-08-25 | lecture-01 | What is a linear system? And what does it mean to ‘solve’ a linear system? |
| 2025-08-27 | lecture-02 | What do the solutions to a linear system look like? |
| 2025-08-29 | lecture-03 | How can we solve a linear system without using subsitution? |
| 2025-09-03 | lecture-04 | What is Gauss-Jordan elimination (AKA row reduction) and how can we use it to solve linear systems? |
| 2025-09-05 | lecture-05 | What is a matrix, and what are the fundamental algebraic operations we can do with matrices? |
| 2025-09-08 | lecture-06 | How do we encode a linear system using matrices? And once thus encoded, what can we say about the linear system just by looking at the matrix? |
| 2025-09-10 | lecture-07 | What is a determinant, geometrically? |
| 2025-09-12 | lecture-08 | What are the key properties of the matrix inverse? |
| 2025-09-15 | lecture-09 | How do we understand (and compute) the determinant, algebraically? |
| 2025-09-17 | lecture-10 | What are some useful connections between the geometric and algebraic interpretations of the determinant? |
| 2025-09-19 | lecture-11 | What are the essential properties of vectors? |
| 2025-09-22 | lecture-12 | What is a vector space? |
| 2025-09-24 | lecture-13 | What do linear subspaces look like? |
| 2025-09-26 | lecture-14 | How do we build a linear subspace from a set of vectors? |
| 2025-09-29 | lecture-15 | How do we know if a linear system is minimal—i.e., that it doesn’t have any redundant equations? |
| 2025-10-01 | lecture-16 | How do we build a vector space from a minimal set of vectors? |
| 2025-10-03 | lecture-17 | Bases and dimension of a vector space |
| 2025-10-06 | lecture-18 | What are the three fundamental subspaces associated with a matrix A? |
| 2025-10-08 | lecture-19 | Examples of calculations of dimension and basis of fundamental subspaces |
| 2025-10-10 | lecture-20 | What is a linear transformation? |
| 2025-10-13 | lecture-21 | Is a basis all you need? |
| 2025-10-15 | lecture-22 | Why do the row space and column space always have the same dimension? |
| 2025-10-17 | lecture-23 | Why is matrix multiplication defined the way it is? |
| 2025-10-20 | lecture-24 | How do we find a matrix to represent a linear transformation? |
| 2025-10-22 | lecture-25 | The matrix we use to represent a linear transformation depends a choice of bases. How does it change when whe choose different bases? |
| 2025-10-24 | lecture-26 | What do linear transformations of 2-dimensional space look like? |
| 2025-10-27 | lecture-27 | What do linear transformations of 2-dimensional space look like? |
| 2025-10-29 | lecture-28 | What are eigenvalues and eigenvectors and how do we find them? |
| 2025-10-31 | lecture-29 | How to compute eigenvectors and eigenvalues? |
| 2025-11-03 | lecture-30 | What can we say about two matrices that represent the same linear transformation? |
| 2025-11-05 | lecture-31 | When is a matrix similar to a diagonal matrix? |
| 2025-11-07 | lecture-32 | How can we test whether a matrix is diagonalizable? |
| 2025-11-10 | lecture-33 | Jordan canonical form |
| 2025-11-12 | lecture-34 | Multiplicity and an application of diagonalizability |
| 2025-11-14 | lecture-35 | What are the eigenvectors of a rotation? |
| 2025-11-17 | lecture-36 | Examples of systems of first order linear differential equations |
| 2025-11-19 | lecture-37 | Examples of systems of first order linear differential equations |
| 2025-11-21 | none | exam review |
| 2025-11-26 | lecture-38 | The matrix exponential |
| 2025-12-01 | lecture-39 | Sections 6.1, 6.2 |
| 2025-12-03 | lecture-40 | An application to biology: the Jukes Cantor Model |
| 2025-12-05 | lecture-41 | What do solutions to Y’=AY look like? |
