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 GFDL or the CC BY-SA 2.5)
A First Course in Linear Algebra by Robert A. Beezer (freely available under the terms of the GFDL)

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 Midterm The exam will cover all materials through HW5. Remember, you may bring one standard 8.5x11 sheet of paper with handwritten notes. 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 Final Exam The exam will cover all materials in HW6 through HW11. Remember, you may bring one standard 8.5x11 sheet of paper with handwritten notes. Location & Time B131 Hamerschlag Hall 4:00–7:00 p.m.

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.

• Notes on Zorn's Lemma and the Axiom of Choice can be found here, courtesy of Boris Bukh.
• A proof that Zorn's Lemma implies that every vector space has a basis can be found here.
• Andreas Blass's paper shows that if every vector space has a basis, then the Axiom of Choice is true.