MA2130: Basic Graph Theory
Planar Graphs
Dr. S. Mishra, Department of Mathematics, I.I.T. Madras
November 7, 2016
Planar Graphs
Definition : A graph G = (V , E) is called a planar graph if it
can be drawn in the plane in such a way that every pair of
edge
Matching Theory
Matching:
Let G = (V , E) be a simple graph. A subset M of E is called a
matching in G if no pair of edges e1 and e2 of M share a
common vertex.
Matching Theory
Matching:
Let G = (V , E) be a simple graph. A subset M of E is called a
match
Indian Institute of Technology Madras
Department of Mathematics
MA 2130 Basic Graph Theory
Assignment-5
1. Write a proof of the fact that every tree is planar.
2. Draw a planar graph in which every vertex has degree exactly 5.
3. Suppose that e is a bridg
Module 3
1
More on Vector Spaces and Linear transformations
1.1 Let X be a vector space over the field F and given a set I, let us write
x
X I := I
X is a f unction.
For x X I , we shall write x(i) X as xi X at each i I. For x, u X I , F, define
x + u X
Module 7
1
Generalized Inverse
1. Theorem Given an m n matrix A over C, there exist unitary matrices U and V of order m and n
respectively such that A = U DV where
(i)
D = diag[1 , . . . , r , 0, 0, . . . , 0] if m = n
(ii)
. . . (1)
1
D=
.
0
0
.
r
.
.
0
Module 6
1
Inner Product Space
1. Recall that X is called a vector space over a field F when X is an abelian group equipped with an
(x,y)x
action X F X satisfying expected identities like (x + u) = x + u, x( + ) = x + x
etc; we also can, and do, write x X
Module 6
1
Orthogonal and Orthonormal Basis
s
1. Let X be a vector space over K, K = R or C. We note that R C and the conjugation. Let XX
K
be a sesquilinear form which is hermitian; that is, s(x, u) = s(u, x). Then s(x, u) + s(x, u) =
s(x, u) + s(u, x)
Module 4
1
Eigenvalues and Eigenvectors
A
Given a linear operator X
X, a scalar F is called an eigen value of A iff the equation Ax = x
has a non-zero solutions; put another way, ker(A I) := cfw_x X | Ax = x is a non-zero subspace
of X. Thus for dim X =
Module 5
1
Sesqui/Bi-Linear forms
Sright
tr
1 Suppose X is right K-module so that X = X (
= X tr is a right K-module again. Let X X
be a right K-linear transformation. Thus to each x X we associate a left K-linear transformation
sright (x)
X K. Recording
Module 4
1
Diagonalization
Now suppose in A () = ( 1 )p1 . . . ( k )pk we have pj = 1 for 1 j k (which means then there
are only linear factors in the minimal polynomial) then there are exactly k distinct eigenvalues, with nj
linearly independent eigenvec
Module 5
1
More on Sesqui/Bi-Linear forms
s
1. Consider now a sesquiliner form X X
K and introduce s (x, u) := s(u, x). Then s is
again a sesquilinear form (s (x, u) = s(u, x) = s(u, x) = s(u, x) = s (x, u)
Let Y CenK; this means Y = Y for each K and he
Classnotes - MA2030
Linear Algebra
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
This is a modified version of the classnotes of A. V. Jayanthan and A. Singh
Contents
1
2
3
4
5
Vector Spaces
1.1 What is a vector space?
1.2
MH2814 Probability & Statistics
Discrete Probability Distribution
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
S
MH2814 Probability & Statistics
Basic Probability Theory
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
SchoolProb
MH2814 Probability & Statistics
Conditional Probability & Independent Events
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Scienc
MH2814 Probability & Statistics
Joint Probability Distribution
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
Scho
Nanyang Technological University
SPMS/Division of Mathematical Sciences
MH2814 Probability & Statistics
Tutorial 7
.
Topics: Geometric distribution and Poisson distribution; Poisson Approximation.
.
1. The probability that a student pilot passes the writt
Nanyang Technological University
SPMS/Division of Mathematical Sciences
2016/2017 Semester 1
MH2814 Probability & Statistics
Assignment 1
Name &
Matriculation Number
Score:
IMPORTANT NOTE
1. Each assignment is allocated 10 points. If you do not score full
MH2814 Probability & Statistics
Continuous Probability Distribution
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
MH2814 Probability & Statistics
Population & Samples
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
SchoolProbabil
Nanyang Technological University
SPMS/Division of Mathematical Sciences
2016/2017 Semester 1
MH2814 Probability & Statistics
Assignment 2
Name &
Matriculation Number
Score:
IMPORTANT NOTE
Submission week: 29th August 2016 -2nd September 2016. (Week for Tu
MH2814 Probability & Statistics
Sample Space
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
SchoolProbability
of P
Nanyang Technological University
SPMS/Division of Mathematical Sciences
MH2814 Probability & Statistics
Tutorial 3
.
Topics: Bayes Rule. Random variables (Discrete , Continuous), Discrete Probability
distribution functions, Cumulative distribution functio
Nanyang Technological University
SPMS/Division of Mathematical Sciences
MH2814 Probability & Statistics
Tutorial 1
.
1. A construction company has five categories of employees as follows:
Category:
1
2
3
4
5
Salary ($ /month) : 600 700 800 900 1000
No. of
Nanyang Technological University
SPMS/Division of Mathematical Sciences
MH2814 Probability & Statistics
Tutorial 1
.
MH2814 Assessments:
1. Final Examination (60%): 2-Hour.
2. Common Tests (30%): TWO 40-min common tests, each contributes 15%.
Dates: 16/9/
Selected Solutions
Math 420
Homework 2
1/20/12
1.5.8 Let
C11 C12
C=
C21 C22
be a 2 2 matrix. We inquire when it is possible to find 2 2 matrices A and B such that
C = AB BA. Prove that such matrices can be found if and only if C11 + C22 = 0.
It is easy to
Nanyang Technological University
SPMS/Division of Mathematical Sciences
MH2814 Probability & Statistics
Tutorial 7
.
Topics: Geometric Distribution and Poisson distribution; Poisson Approximation.
.
1. The probability that a student pilot passes the writt
MH2814 Probability & Statistics
Expectation & Variance
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathematical Sciences
MH2814
SchoolProbab