21-241: Matrices and Linear Transformations
Course Information
- Lecture
- 4709 Wean Hall
- MTWRF 10:30–11:50 a.m.
- Office hours
- 6201 Wean Hall
- MWF 12:00–2:00 p.m.
- Textbooks
- Linear Algebra by Jim Hefferon (freely available under the terms of the GFDLGNU Free Documentation License or the CC BY-SA 2.5Creative Commons Attribution-ShareAlike 2.5 Generic)
- A First Course in Linear Algebra by Robert A. Beezer (freely available under the terms of the GFDLGNU Free Documentation License)
Schedule and Assignments
Homework Template
Homework assignments are due on the date by which they are posted.
Solutions will be posted after the lecture that day.
| Week #1 | ||
| July 1 | Systems of linear equations, Gaussian elimination, matrix-vector representations | |
| July 2 | Matrix rank, homogenous and non-homogeneous solutions | HW1 | Solutions |
| July 3 | Matrix operations, matrix inverse | |
| July 4 | No Class (Independence Day) | |
| July 5 | Matrix inverse (cont), elementary matrices, DSW1 | HW2 | Solutions |
| Week #2 | ||
| July 8 | Vector spaces, subpaces | |
| July 9 | Subspace properties, principle of mathematical induction, span | HW3 | Solutions |
| July 10 | Linear independence, basis, dimension | |
| July 11 | Four fundamental subspaces of a matrix, matrix rank for realsies (or complexsies) | |
| July 12 | Fisher's theorem, rank–nullity theorem, DSW2 | HW4 | Solutions |
| Week #3 | ||
| July 15 | Inner products, norms, Cauchy–Schwarz inequality | |
| July 16 | Orthonormal bases, Gram–Schmidt algorithm, orthogonal complement | HW5 | Solutions |
| July 17 |
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| July 18 | Orthogonal complements (cont), pathological examples in infinitely many dimensions, projections | |
| July 19 | Linear transformations, every reasonable function is a matrix, DSW3 | HW6 | Solutions |
| Week #4 | ||
| July 22 | Linear transformations (cont), isomorphisms, partial converse to the rank–nullity theorem | |
| July 23 | Orthogonal matrices, isometries of $\mathbb{R}^n$ | HW7 | Solutions |
| July 24 | Change-of-basis, similarity, diagonalizable matrices | |
| July 25 | A first look at eigenvalues and eigenvectors, Schur's triangularization theorem | |
| July 26 | Determinants, DSW4 | HW8 | Solutions |
| Week #5 | ||
| July 29 | Determinants (cont) | |
| July 30 | Characteristic polynomials, eigenvalues and eigenvectors, spectral theorem | HW9 | Solutions |
| July 31 | Cayley–Hamilton theorem, Fibonacci numbers | No office hours today |
| Aug 1 | Positive (semi)definite matrices | Office hours from 12–2 |
| Aug 2 | Invariant subpaces, DSW5 | HW10 | Solutions |
| Week #6 | ||
| Aug 5 | Invariant subspaces (cont), Jordan canonical form | |
| Aug 6 | Jordan canonical form (cont) | |
| Aug 7 | Funzies: almost orthogonal vectors, decomposing rectangles into squares, etc | HW11 | Solutions |
| Aug 8 | Funzies: Hadamard's theorem, etc, DSW6 | |
| Aug 9 |
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Supplementary Materials
It turns out that every vector space having a basis is equivalent to the Axiom of Choice. This material is beyond the scope of this course, but I still recommend giving it a read.