21241 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 BYSA 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, matrixvector representations  
July 2  Matrix rank, homogenous and nonhomogeneous 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 


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  Changeofbasis, 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 


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.