REU 2013: Apprentice Program
Summer 2013
Lecture 1: June 24, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko
Small Digression to Graph Theory
Exercise 1.1 (Puzzle) Number of people who have made an odd number of handshakes must be
even.
Denition 1.2 A graph G
REU 2013: Apprentice Program
Summer 2013
Lecture 5: July 2, 2013
Madhur Tulsiani
1
Scribe: David Kim
Determinants continued
Recall the denition of the determinant,
n
sgn( )
det(A) =
Sn
Ai,(i)
i=1
where each : [n] [n] is a permutation and A Mn (F ).
Claim
REU 2013: Apprentice Program
Summer 2013
Lecture 05: July 1st, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko
Linear Transformation continued
Recall that : V W with e = cfw_e1 , . . . , en as a basis for V and f = cfw_f1 , . . . , fm as basis for W .
Then
REU 2013: Apprentice Program
Summer 2013
Lecture 3: June 27, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko
Mapping between vector spaces
1.1
Basic denitions
Denition 1.1 (Linear Map) We call a mapping from vector space V to W a linear map if
1. v1 , v2 V, (
REU 2013: Apprentice Program
Summer 2013
Lecture 2: June 25, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko
Lagrange Interpolation continued
Last class Lagrange interpolation was left as a homework exercise. In fact you can explicitly
calculate the coecients
REU 2013: Apprentice Program
Summer 2013
Lecture 6: July 5, 2013
Madhur Tulsiani
1
Scribe: David Kim
Eigenvalues & Eigenvectors
Denition 1.1 (Eigenvalue & Eigenvector) Let A Cnn . Then C is said to be an
eigenvalue of A if v = 0 such that
Av = v (A I )v =
REU 2013: Apprentice Program
Summer 2013
Lecture 7: July 8, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko, David Kim
Gram-Schmidt Orthononalization
Exercise 1.1 If U is unitary and is a (complex) eigenvalue, prove that | = 1.
Recall that b1 , . . . , bk for
UChicago REU 2013 Apprentice Program
Spring 2013
Lecture 12: July 17th, 2013
Prof. Babai
Scribe: David Kim
The students asked the following questions: #69 (Annie), Oddtown Problem (Nate), #78 (Annie,
Philip), #82(3) (Freddy), and (*) = 0 has multiplicity
UChicago REU 2013 Apprentice Program
Summer 2013
Lecture 10: July 15, 2013
Instructor: Madhur Tulsiani
1
Scribe: David Kim
Eigenvalues of the adjacency matrix
All graphs in this lecture will be undirected.
Let G = (V, E ) be a graph with n vertices. Let A
REU 2013: Apprentice Program
Summer 2013
Lecture 9: July 11, 2013
Madhur Tulsiani
1
Scribe: Young Kun Ko, David Kim
More on Adjacency Matrices
Recall that we have G = (V, E ) and its adjacency matrix A and eigenvalues 1 2 . . . n
and
i = tr(A) = 0.
Exerci
REU 2013: Apprentice Program
Summer 2013
Lecture 8: July 9, 2013
Madhur Tulsiani
1
Scribe: David Kim, Young Kun Ko
Eigenvalues and eigenvectors of adjacency matrices
Recall given an adjacency matrix for an undirected graph G = (V, E ), A is real and symme