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.