MA MSc Mathematics Sem I II1 Syllabus Mumbai University by munotes
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UNIVERSITY OF MUMBAI
Proposed Syllabus for
M.A./M.Sc. Semester I & II (CBCS)
Program: M.A/M.Sc.
Course: Mathematics
with eect from the academic year 2017-2018
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M.A./M.Sc. Semester I and II
Choice Based Credit System (CBCS)
Semester I
Algebra I
Course Code Unit Topics Credits L/W
PSMT101,PAMT101Unit I Dual spaces
Unit II Determinants 5 4
Unit III Characteristic polynomial
Unit IV Bilinear forms
Analysis I
Course Code Unit Topics Credits L/W
PSMT102,PAMT102Unit I Euclidean space Rn
Unit II Riemann integration 5 4
Unit III Dierential functions
Unit IV Inverse function theorem , Implicit
Unit IV function theorem
Complex Analysis
Course Code Unit Topics Credits L/W
PSMT103,PAMT103Unit I Holomorphic functions
Unit II Contour integration, Cauchy-Gursat theorem 5 4
Unit III Properties of holomorphic functions
Unit IV Conformal mappings, Residue Calculus
Discrete Mathematics
Course Code Unit Topics Credits L/W
PSMT104,PAMT104Unit I Number Theory
Unit II Advanced Counting 5 4
Unit III Recurrence relations
Unit IV Polya's theory of counting
Set Theory and Logic
Course Code Unit Topics Credits L/W
PSMT105,PAMT105Unit I Introduction to logic
Unit II Sets and functions 4 4
Unit III Partial order
Unit IV Lattices
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Semester II
Algebra II
Course Code Unit Topics Credits L/W
PSMT201,PAMT201Unit I Groups, group homomorphisms
Unit II Groups acting on sets and Syllow theorems 5 4
Unit III Rings, elds
Unit IV Divisibility in integral domains
Topology
Course Code Unit Topics Credits L/W
PSMT202,PAMT202Unit I Topological spaces
Unit II Connected Topological spaces 5 4
Unit III Compact Topological spaces
Compact metric spaces, Complete
Unit IV metric spaces
Analysis II
Course Code Unit Topics Credits L/W
PSMT203,PAMT203Unit I Measures
Unit II measurable functions and 5 4
integration of non-negative functions
Unit III Dominated convergence theorem
andL1;L2spaces.
Unit IV Signed measures, Radon-Nykodym theorem
Dierential Equations
Course Code Unit Topics Credits L/W
PSMT204,PAMT204Unit I Picard's theorem
Unit II Ordinary dierential equations 5 4
Unit III Sturm-Liouville theory
Unit IV First order Partial
Dierential Equations
Probability Theory
Course Code Unit Topics Credits L/W
PSMT205,PAMT205Unit I Basics of Probability
Unit II Probability measure 4 4
Unit III Random variables
Unit IV Limit theorems
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Teaching Pattern for Semester I and II
1. Four lectures per week per course. Each lecture is of 60 minutes duration.
2. In addition, there shall be tutorials, seminars as necessary for each of the ve courses.
SEMESTER I
All Results have to be done with proof unless otherwise stated.
PSMT101,PAMT101 ALGEBRA I
Unit I. Dual spaces (15 Lectures)
Para 1 and 2 of Unit Iare to be reviewed without proof (no question be asked).
1. Vector spaces over a eld, linear independence, basis for nite dimensional and innite
dimensional vector spaces and dimension.
2. Kernel and image, rank and nullity of a linear transformation, rank-nullity theorem (for nite
dimensional vector spaces), relationship of linear transformations with matrices, invertible linear
transformations. The following are equivalent for a linear map T:V !Vof a nite dimensional
vector space V:
1.Tis an isomorphism.
2. kerT=f0g:
3. Im (T) =V:
3. Linear functionals, dual spaces of a vector space, dual basis (for nite dimensional vector
spaces), annihilator Win the dual space Vof a subspace Wof a vector space Vand dimension
formula, ak dimensional subspace of an n dimensional vector space is intersection of n k
many hyperspaces. Double dual Vof a Vector space Vand canonical embedding of Vinto
V:Vis isomorphic to VwhenVis of nite dimension. (ref:[1] Hoffman K and Kunze
R)
4. Transpose Ttof a linear transformation T:For nite dimensional vector spaces: rank
(Tt)=rankT;range(Tt) is the annihilator of kernel ( T), matrix representing Tta rank of a
matrix. (ref:[1] Hoffman K and Kunze R )
Unit II. Determinants (15 Lectures)
Determinants as alternating n-forms, existence and uniqueness , Laplace expansion of determi-
nant, determinants of products and transposes, determinants and invertible linear transforma-
tions, determinant of a linear transformation.
Reference for Unit II: [1] Hoffman K and Kunze R ,Linear Algebra .
Unit III. Characteristic polynomial (15 Lectures)
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Eigen values and Eigen vectors of a linear transformation, Characteristic polynomial, Cayley-
Hamilton theorem, Minimal polynomial, Triangulable and diagonalizable linear operators, invari-
ant subspaces and simple matrix representation (for nite dimension). (ref: [5] N.S. Gopalkr-
ishnan & [3] Serge Lang )
Nilpotent linear transformations on nite dimensional vector spaces, index of a Nilpotent linear
transformation. Linear independence of fu;Nu;;Nk 1ugwhereNis a nilpotent linear
transformation of index k2of a vector space Vandu2VwithNu6= 0:(Ref: [2]
I.N.Herstein )
For a nilpotent linear transformation Nof a nite dimensional vector space Vand for any
subspaceWofVwhich is invariant under N;there exists a subspace V1ofVsuch that
V=WV1:(Ref:[2] I.N.Herstein )
Computations of Minimum polynomials and Jordan Canonical Forms for 33-matrices through
examples of matrices such as0
@3 1 1
2 2 1
2 2 01
A;0
@1 1 1